Digital communication using lattice division multiplexing

ABSTRACT

A wireless data transmission technique includes encoding information bits as a periodic sequence of quadrature amplitude modulation (QAM) symbols, convolving the periodic sequence with a periodic pulse function, thereby generating a filtered periodic sequence, transforming the filtered periodic sequence to a delay-Doppler domain waveform, converting the delay-Doppler domain waveform to a time domain waveform, and transmitting the time domain waveform.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent document is a 371 National Phase Application of PCT Application No. PCT/US2018/029209 entitled “DIGITAL COMMUNICATION USING LATTICE DIVISION MULTIPLEXING” filed on Apr. 24, 2018 which claims priority to and benefits of U.S. Provisional Patent Application No. 62/489,401 entitled “DIGITAL COMMUNICATION USING LATTICE DIVISION MULTIPLEXING” filed on Apr. 24, 2017. The entire contents of the aforementioned patent applications are incorporated by reference as part of the disclosure of this patent document.

TECHNICAL FIELD

The present document relates to wireless communication, and more particularly, to data modulations schemes used in wireless communication.

BACKGROUND

Due to an explosive growth in the number of wireless user devices and the amount of wireless data that these devices can generate or consume, current wireless communication networks are fast running out of bandwidth to accommodate such a high growth in data traffic and provide high quality of service to users.

Various efforts are underway in the telecommunication industry to come up with next generation of wireless technologies that can keep up with the demand on performance of wireless devices and networks.

SUMMARY

This document discloses techniques that can be used to implement transmitters and receivers for communicating using a modulation technique called lattice division multiplexing.

In one example aspect, wireless communication method, implementable by a wireless communication apparatus is disclosed. The method includes encoding information bits as a periodic sequence of quadrature amplitude modulation (QAM) symbols, convolving the periodic sequence with a periodic pulse function, thereby generating a filtered periodic sequence, transforming the filtered periodic sequence to a delay-Doppler domain waveform, converting the delay-Doppler domain waveform to a time domain waveform, and transmitting the time domain waveform.

In another aspect, a wireless communication method that includes transforming an information signal to a discrete lattice domain signal, shaping bandwidth and duration of the discrete lattice domain signal by a two-dimensional filtering procedure to generate a filtered information signal, generating a time domain signal from the filtered information signal, and transmitting the time domain signal over a wireless communication channel is disclosed.

In another example aspect, a wireless communication apparatus that implements the above-described methods is disclosed.

In yet another example aspect, the method may be embodied as processor-executable code and may be stored on a computer-readable program medium.

These, and other, features are described in this document.

BRIEF DESCRIPTION OF THE DRAWINGS

Drawings described herein are used to provide a further understanding and constitute a part of this application. Example embodiments and illustrations thereof are used to explain the technology rather than limiting its scope.

FIG. 1 shows an example of a wireless communication system.

FIG. 2 pictorially depicts relationship between time, frequency and Zak domains.

FIG. 3 pictorially depicts the periodic and quasi-periodic nature of an information grid in the Zak domain.

FIG. 4 pictorially depicts the transformation of a single Quadrature Amplitude Modulation (QAM) symbol.

FIG. 5 pictorially depicts periodicity of a QAM symbol.

FIG. 6 is a graphical representation of an OTFS waveform.

FIG. 7 is a graphical representation of a windowing function.

FIG. 8 is a graphical comparison of transmit waveforms of a single QAM symbol using OTFS and OTFS-MC (multicarrier).

FIG. 9 shows a combined waveform for multiple symbols.

FIG. 10 shows a waveform of modulated QAM signals.

FIG. 11 is a flowchart representation of an example of a wireless communication method.

FIG. 12 is a flowchart representation of another example of a wireless communication method.

FIG. 13 shows an example of a wireless transceiver apparatus.

FIG. 14 shows an example of a hexagonal lattice. The hexagon at the center region encloses Voronoi region around the zero lattice point. The two lattice points with arrow decoration are the basis of the maximal rectangular sub-lattice.

FIG. 15 pictorially depicts an example in which the Zak domain and the time/frequency Zak transforms realizing the signal space realization lying in between time and frequency realizations.

FIG. 16 shows a depiction of the Zak to generalized Zak intertwining transformation.

DETAILED DESCRIPTION

To make the purposes, technical solutions and advantages of this disclosure more apparent, various embodiments are described in detail below with reference to the drawings. Unless otherwise noted, embodiments and features in embodiments of the present document may be combined with each other.

Section headings are used in the present document to improve readability of the description and do not in any way limit the discussion of techniques or the embodiments to the respective sections only.

Traditional multi-carrier (MC) transmissions schemes such as orthogonal frequency division multiplexing (OFDM) schemes are characterized by two parameters: symbol period (or repetition rate) and subcarrier spacing. The symbols include a cyclic prefix (CP), whose size typically depends on the delay of the wireless channel for which the OFDM modulation scheme is being used. In other words, CP size is often fixed based on channel delay and if symbols are shrunk to increase system rate, it simply results in the CP becoming a greater and greater overhead. Furthermore, closely placed subcarriers can cause inter-carrier interference and thus OFDM systems have a practical limit on how close the subcarriers can be placed to each other without causing unacceptable level of interference, which makes it harder for a receiver to successfully receive the transmitted data.

The theoretical framework disclosed in the present document, including Appendix A and Appendix B, can be used to build signal transmission and reception equipment that can overcome the above discussed problems, among others.

This patent document discloses, among other techniques, a lattice division multiplexing technique that, in some embodiments, can be used to implement embodiments that can perform multi-carrier digital communication without having to rely on CP.

For the sake of illustration, many embodiments disclosed herein are described with reference to the Zak transform. However, one of skill in the art will understand that other transforms with similar mathematical properties may also be used by implementations. For example, such transforms may include transforms that can be represented as an infinite series in which each term is a product of a dilation of a translation by an integer of the function and an exponential function.

FIG. 1 shows an example communication network 100 in which the disclosed technologies can be implemented. The network 100 may include a base station transmitter that transmits wireless signals s(t) (e.g., downlink signals) to one or more receivers 102, the received signal being denoted as r(t), which may be located in a variety of locations, including inside or outside a building and in a moving vehicle. The receivers may transmit uplink transmissions to the base station, typically located near the wireless transmitter. The technology described herein may be implemented at a receiver 102 or at the transmitter (e.g., a base station).

Signal transmissions in a wireless network may be represented by describing the waveforms in the time domain, in the frequency domain, or in the delay-Doppler domain (e.g., Zak domain). Because these three represent three different ways of describing the signals, signal in one domain can be converted into signal in the other domain via a transform. For example, a time-Zak transform may be used to convert from Zak domain to time domain. For example, a frequency-Zak transform may be used to convert from the Zak domain to the frequency domain. For example, the Fourier transform (or its inverse) may be used to convert between the time and frequency domains.

FIG. 2 shows the relationships among the various domains. The horizontal axis represents time, the vertical axis represents frequency and signal in the Zak domain is shown as a 2-D shape in the (τ, ν), or delay-Doppler domain.

A Zak waveform could be represented as a function of two variables—delay (τ) and Doppler (ν). As disclosed later on in this document, a Zak waveform is periodic along the Doppler variable and quasi-periodic (up to a phase term) in the delay variable.

FIG. 3 pictorially depicts the periodic nature of a Zak waveform in the two dimensions delay (τ) and Doppler (ν), where each solid circle represents a QAM symbol based on the data being carried in the Zak waveform.

FIG. 4 depicts a lattice domain representation of a QAM symbol 402. The lattice domain is a transform domain representation using two dimensions represented by two independent vectors (e.g., delay and Doppler or time-frequency). While not explicitly shown for the sake of clarity, it is understood that the QAM symbol will mathematically occur periodically in the vertical dimension in FIG. 4. Furthermore, while also not shown for the sake of clarity, during operation, multiple QAM symbols will be mapped in the rectangle 404. Additionally, the QAM symbol is quasi-periodic in the horizontal direction (representing time), and the corresponding phases in each instance are shown by the equations below the graph.

FIG. 5 graphically shows the effect of integration along the ν dimension. For example, as shown by the equation 502:

Z(x) = ∫₀^(v_(τ))x(τ, v) d v

Z(x) represents the Zak domain to time transform, representing integration along the Doppler domain ν.

FIG. 6 is a graphical representation of an Orthogonal Time Frequency Space (OTFS) waveform 602 plotted along the delay dimension (horizontal) and Doppler dimension (vertical axis).

FIG. 7 is a graphical representation of a windowing function 702 using same coordinate systems as in FIG. 6. The windowing function 702 operates along time and is depicted along the delay dimension and stretches over one period of the OTFS waveform 602.

FIG. 8 is a graphical comparison of transmit waveforms of a single QAM symbol using OTFS and OTFS-MC (multicarrier).

FIG. 9 shows a combined waveform for multiple symbols, depicting underlying impulses and shaped pulses, as further explained herein.

FIG. 10 shows a waveform of modulated QAM symbols along time (horizontal axis) direction.

The sections numbered “A” and “B” below provide additional mathematical properties and practical uses of the signal waveforms and graphs depicted in FIGS. 5 to 10.

A0. Introduction to OTFS Modulation from Zak Theoretic Point

Next few sections explain the OTFS modulation from the Zak theoretic point of view. This line of exposition push to the forefront the independent status of OTFS as a novel modulation technique and reveals its unique mathematical attributes. This, in contrast to the alternative approach of presenting OTFS as a preprocessing step over MC modulation which somehow obscures the true nature of OTFS and also sacrifice some of its unique strengths. We focus our attention on the following core theoretical topics:

(1) Heisenberg theory.

(2) Zak theory.

(3) OTFS modulation.

(4) Symplectic Fourier duality relation between OTFS and Multi Carrier modulations which is a particular case of the general relation between Radar theory and communication theory.

Before proceeding into a detailed development, it is beneficial to give a brief outline. In signal processing, it is traditional to represent signals (or waveforms) either in time or in the frequency domain. Each representation reveals different attributes of the signal. The dictionary between these two realizations is the Fourier transform: FT:L ₂(t∈

)→L ₂(f∈

),  (0.1)

Interestingly, there is another domain where signals can be naturally realized. This domain is called the delay Doppler domain. For the purpose of the present discussion, this is also referred to as the Zak domain. In its simplest form, a Zak signal is a function φ(τ,ν) of two variables. The variable τ is called delay and the variable ν is called Doppler. The function φ(τ,ν) is assumed to be periodic along ν with period ν_(r) and quasi-periodic along τ with period τ_(r). The quasi periodicity condition is given by: φ(τ+nτ _(r) ,+mν _(r))=exp(j2πnν·τ _(r))φ(τ,ν),  (0.2)

for every n, m∈

. The periods are assumed to satisfy the Nyquist condition τ_(r)·ν_(r)=1. Zak domain signals are related to time and frequency domain signals through canonical transforms Z_(t) and Z_(f) called the time and frequency Zak transforms. In more precise terms, denoting the Hilbert space of Zak signals by H_(z), the time and frequency Zak transforms are linear transformations: Z _(t) :H _(z) →L ₂(t∈

),  (0.3) Z _(f) :H _(z) →L ₂(f∈

).  (0.4)

The pair Z_(t) and Z_(f) establishes a factorization of the Fourier transform FT=Z_(t)·[Z_(f)]⁻¹. This factorization is sometimes referred to as the Zak factorization. The Zak factorization embodies the combinatorics of the fast Fourier transform algorithm. The precise formulas for the Zak transforms will be given in the sequel. At this point it is enough to say that they are principally geometric projections: the time Zak transform is integration along the Doppler variable and reciprocally the frequency Zak transform is integration along the delay variable. The different signal domains and the transformations connecting between them are depicted in FIG. 2.

We next proceed to give the outline of the OTFS modulation. The key thing to note is that the Zak transform plays for OTFS the same role the Fourier transform plays for OFDM. More specifically, in OTFS, the information bits are encoded on the delay Doppler domain as a Zak signal x(τ,ν) and transmitted through the rule: OTFS(x)=Z _(t)(w* _(σ) x(τ,ν)),  (0.5)

where w*_(σ)x(τ,ν) stands for two-dimensional filtering operation with a 2D pulse w(τ,ν) using an operation *_(σ) called twisted convolution (to be explained in the present document). The conversion to the physical time domain is done using the Zak transform. Formula (0.5) should be contrasted with the analogue formulas in case of frequency division multiple access FDMA and time division multiple access TDMA. In FDMA, the information bits are encoded on the frequency domain as a signal x(f) and transmitted through the rule: FDMA(x)=FT(w(f)*x(f)),  (0.6)

where the filtering is done on the frequency domain by linear convolution with a 1D pulse w(f) (in case of standard OFDM w(f) is equal an sinc function). The modulation mapping is the Fourier transform. In TDMA, the information bits are encoded on the time domain as a signal x(t) and transmitted through the rule: TDMA(x)=Id(w(t)*x(t)),  (0.7)

where the filtering is done on the time domain by linear convolution with a 1D pulse w(t). The modulation mapping in this case is identity.

A1. Heisenberg Theory

In this section we introduce the Heisenberg group and the associated Heisenberg representation. These constitute the fundamental structures of signal processing. In a nutshell, signal processing can be cast as the study of various realizations of signals under Heisenberg operations of delay and phase modulation.

A1.1 The Delay Doppler plane. The most fundamental structure is the delay Doppler plane V=

² equipped with the standard symplectic form: ω(ν₁,ν₂)=ν₁τ₂−τ₁ν₂,  (1.1)

for every ν₁=(τ₁,ν₁) and ν₂=(τ₂, ν₂). Another way to express ω is to arrange the vectors ν₁ and ν₂ as the columns of a 2×2 matrix so that ω (ν₁,ν₂) is equal the additive inverse of the matrix determinant.

${\omega\left( {v_{1},v_{2}} \right)} = {- {{\det\begin{bmatrix} | & | \\ v_{1} & v_{2} \\ | & | \end{bmatrix}}.}}$

The symplectic form is anti-symmetric ω (ν₁, ν₂)=−ω(ν₂,ν₁), thus, in particular ω (ν,ν)=0 for every ν∈V. We also consider the polarization form: β(ν₁,ν₂)=ν₁τ₂,  (1.2)

for every ν₁=(τ₁,ν₁) and ν₂=(τ₂, ν₂). We have that: β(ν₁,ν₂)−β(ν₂,ν₁)=ω(ν₁,ν₂),  (1.3)

The form β should be thought of as “half” of the symplectic form. Finally, we denote by ψ(z)=exp(2πiz) is the standard one-dimensional Fourier exponent.

A1.2 The Heisenberg group. The polarization form β gives rise to a two step unipotent group called the Heisenberg group. As a set, the Heisenberg group is realized as Heis=V×S¹ where the multiplication rule is given by: (ν₁ ,z ₁)·(ν₂ ,z ₂)=(ν₁+ν₂,exp(j2πβ(ν₁,ν₂))z ₁ z ₂).  (1.4)

One can verify that indeed rule (1.4) yields a group structure: it is associative, the element (0, 1) acts as unit and the inverse of the element (ν, z) is given by: (ν,z)⁻¹=(−ν,exp(j2πβ(ν,ν))z ⁻¹)

Most importantly, the Heisenberg group is not commutative. In general, (ν₁, z₁)(ν₂, z₂)≠(ν₂, z₂)·(ν₁, z₁). The center consists of all elements of the form (0,z), z∈S¹. The multiplication rule gives rise to a group convolution operation between functions:

$\begin{matrix} \begin{matrix} {{h_{1}*_{\sigma}{h_{2}(v)}} = {\int\limits_{{v_{1} + v_{2}} = v}{\exp\mspace{11mu}\left( {j\; 2{\pi\beta}\mspace{11mu}\left( {v_{1},v_{2}} \right)} \right){h_{1}\left( v_{1} \right)}{h_{2}\left( v_{2} \right)}}}} \\ {{= {\int\limits_{v^{\prime}}{\exp\mspace{11mu}\left( {j\; 2{\pi\beta}\mspace{11mu}\left( {v^{\prime},{v - v^{\prime}}} \right)} \right){h_{1}\left( v^{\prime} \right)}{h_{2}\left( {v - v^{\prime}} \right)}d\; v^{\prime}}}},} \end{matrix} & (1.5) \end{matrix}$

for every pair of functions h₁, h₂∈

(V). We refer to (1.5) as Heisenberg convolution or twisted convolution. We note that a twisted convolution differs from linear convolution through the additional phase factor exp (j2πβ(ν₁,ν₂)).

A1.3 The Heisenberg representation The representation theory of the Heisenberg group is relatively simple. In a nutshell, fixing the action of the center, there is a unique (up-to isomorphism) irreducible representation. This uniqueness is referred to as the Stone-von Neumann property. The precise statement is summarized in the following theorem: Theorem 1.1 (Stone-von-Neumann Theorem). There is a unique (up to isomorphism) irreducible Unitary representation π: Heis→U(H) such that π(0,z)=zId_(H).

In concrete terms, the Heisenberg representation is a collection of unitary operators π(ν)∈U(H), for every ν∈V satisfying the multiplicativity relation: π(ν₁)·π(ν₂)=exp(j2πβ(ν₁,ν₂))π(ν₁+ν₂),  (1.6)

for every ν₁, ν₂∈V. In other words, the relations between the various operators in the family are encoded in the structure of the Heisenberg group. An equivalent way to view the Heisenberg representation is as a linear transform Π:

(V)→Op(H), taking a function h∈

(V) and sending it to the operator Π(h)∈Op(H) given by:

$\begin{matrix} {{{\Pi(h)} = {\int\limits_{\upsilon \in V}{{h(v)}d\; v}}},} & (1.7) \end{matrix}$

The multiplicativity relation (1.6) translates to the fact that H interchanges between Heisenberg convolution of functions and composition of linear transformations, i.e., Π(h ₁*_(σ) h ₂)=Π(h ₁)·Π(h ₂),  (1.8)

for every h₁, h₂∈

(V). Interestingly, the representation π, although is unique, admits multitude of realizations. Particularly well known are the time and frequency realizations, both defined on the Hilbert space of complex valued functions on the real line H=L₂(

). For every x∈

, we define two basic unitary transforms: L _(x)(φ)(y)=φ(y−x),  (1.9) M _(x)(φ)(y)=exp(j2πxy)φ(y).  (1.10)

for every φ∈H. The transform L_(x) is called delay by x and the transform M_(x) is called modulation by x. Given a point ν=(τ, V)∈V we define the time realization of the Heisenberg representation by: π_(t)(ν)

φ=L _(τ) ·M _(ν)(φ).  (1.11)

where we use the notation

to designate the application of an operator on a vector. It is usual in this context to denote the basic coordinate function by t (time). Under this convention, the right-hand side of (1.11) takes the explicit form exp (j2πν(t−τ))φ(t−τ). Reciprocally, we define the frequency realization of the Heisenberg representation by: π_(f)(ν)

φ=M− _(τ) ·L _(ν)(φ).  (1.12)

In this context, it is accustom to denote the basic coordinate function by f (frequency). Under this convention, the right-hand side of (1.12) takes the explicit form exp (−j2πτf)φ(f−ν). By Theorem 1.1, the time and frequency realizations are isomorphic in the sense that there is an intertwining transform translating between the time and frequency Heisenberg actions. The intertwining transform in this case is the Fourier transform: FT(φ)(f)=∫_(t) exp(−j2πft)φ(t)dt,  (1.13)

for every φ∈H. The time and frequency Heisenberg operators π_(t)(ν, z) and π_(f) (ν, z) are interchanged via the Fourier transform in the sense that: FT·π _(t)(ν)=π_(f)(ν)·FT,  (1.14)

for every ν∈V. We stress that from the point of view of representation theory the characteristic property of the Fourier transform is the interchanging equation (1.14).

A2. Zak Theory

In this section we describe the Zak realization of the signal space. A Zak realization depends on a choice of a parameter. This parameter is a critically sampled lattice in the delay Doppler plane. Hence, first we devote some time to get some familiarity with the basic theory of lattices. For simplicity, we focus our attention on rectangular lattices. The extension of the theory to non-rectangular lattices (called Heisenberg lattices) appears in the more comprehensive manuscript.

A2.1 Delay Doppler Lattices.

A delay Doppler lattice is an integral span of a pair of linear independent vectors g₁, g₂∈V. In more details, given such a pair, the associated lattice is the set: Λ={a ₁ g ₁ +a ₂ g ₂ :a ₁ ,a ₂∈

}.  (2.1)

The vectors g₁ and g₂ are called the lattice basis vectors. It is convenient to arrange the basis vectors as the first and second columns of a matrix G, i.e.:

$\begin{matrix} {{G = \begin{bmatrix} | & | \\ g_{1} & g_{2} \\ | & | \end{bmatrix}},} & (2.2) \end{matrix}$

referred to as the basis matrix. In this way the lattice Λ=G(

²), that is, the image of the standard lattice under the matrix G. The volume of the lattice is by definition the area of the fundamental domain which is equal to the absolute value of the determinant of G. Every lattice admits a symplectic reciprocal lattice, aka orthogonal complement lattice that we denote by Λ^(⊥). The definition of Λ^(⊥) is: Λ^(⊥) ={ν∈V:ω(ν,λ)∈

for every λ∈Λ}.  (2.3)

We say that A is under-sampled if Λ⊂Λ^(⊥). We say that A is critically sampled if Λ=Λ^(⊥). Alternatively, an under-sampled lattice is such that the volume of its fundamental domain is ≥1. From this point on we consider only under-sampled lattices. Given a lattice Λ, we define its maximal rectangular sub-lattice as Λ_(r)=

τ_(r)⊕

ν_(r) where: τ_(r)=arg min {τ>0:(τ0)∈Λ},  (2.4) ν_(r)=arg min {ν>0:(0,ν)∈Λ}.  (2.5)

When either τ_(r) or ν_(r), are infinite, we define Λ_(r)={0}. We say a lattice Λ is rectangular if Λ=Λ_(r). Evidently, a sub-lattice of a rectangular lattice is also rectangular. A rectangular lattice is under-sampled if τ_(r)ν_(r)≥1. The standard example of a critically sampled rectangular lattice is Λ_(rec)=

⊕

, generated by the unit matrix:

$\begin{matrix} {G_{rec} = {\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.}} & (2.6) \end{matrix}$

An important example of critically sampled lattice that is not rectangular is the hexagonal lattice A_(hex), see FIG. 14, generated by the basis matrix:

$\begin{matrix} {{G_{hex} = \begin{bmatrix} a & {a\text{/}2} \\ 0 & a^{- 1} \end{bmatrix}},} & (2.7) \end{matrix}$

where α=√{square root over (2√{square root over (3)})} The interesting attribute of the hexagonal lattice is that among all critically sampled lattices it has the longest distance between neighboring points. The maximal rectangular sub-lattice of Λ_(hex) is generated by g₁ and 2g₂−g₁, see the two lattice points decorated with arrow heads in FIG. 14. From this point on we consider only rectangular lattices.

A2.2 Zak waveforms A Zak realization is parametrized by a choice of a critically sampled lattice: Λ=

(τ_(r),0)⊕

(0,ν_(r))  (2.8)

where τ_(r)·ν_(r)=1. The signals in a Zak realization are called Zak signals. Fixing the lattice Λ, a Zak signal is a function φ:V→

that satisfies the following quasi periodicity condition: φ(ν+λ)=exp(j2β(ν,λ))φ(ν),  (2.9)

for every ν∈V and λ∈Λ. Writing λ=(kτ_(r),lν_(r)), condition (2.9) takes the concrete form: φ(τ+kτ _(r) +lν _(r))=exp(j2πνkτ _(r))φ(τ,ν),  (2.10)

that is to say that φ is periodic function along the Doppler dimension with period ν_(r) and quasi-periodic function along the delay dimension with quasi period τ_(r). In conclusion, we denote the Hilbert space of Zak signals by H_(z).

A2.3 Heisenberg action The Hilbert space of Zak signals supports a realization of the Heisenberg representation. Given an element u∈V, the corresponding Heisenberg operator π_(z)(U) is given by: (2.11) {π_(z)(u)

φ}(ν)=exp(j2πβ(u,ν−u))φ(ν−u),

for every φ∈H_(z). In words, the element u acts through two-dimensional shift in combination with modulation by a linear phase. The Heisenberg action simplifies in case the element u belongs to the lattice. A direct computation reveals that in this case the action of u=λ∈Λ takes the form: {π_(z)(λ)

φ}(ν)=exp(j2πω(λ,ν))φ(ν),  (2.12)

In words, the operator π_(z)(λ) is multiplication with the symplectic Fourier exponent associated with the point λ. Consequently, the extended action of an impulse function h∈

(V) is given by:

$\begin{matrix} \begin{matrix} {{\left\{ {{\Pi_{z}(h)} \vartriangleright \varphi} \right\}(v)} = {\int\limits_{u \in V}{{h(u)}\left\{ {{\pi_{z}(u)} \vartriangleright \varphi} \right\}(v)\; d\; u}}} \\ {{= {\int\limits_{u \in V}{{\psi\left( {j\; 2\;{\pi\beta}\;\left( {u,{v - u}} \right)} \right)}{h(u)}{\varphi\left( {v - u} \right)}d\; u}}},} \end{matrix} & (2.13) \end{matrix}$

for every φ∈H_(z). In fact, Π_(z)(h)

φ=h*_(σ)φ, that is to say that the extended action is given by twisted convolution of the impulse h with the waveform φ.

A2.4 Zak transforms There are canonical intertwining transforms converting between Zak signals and time/frequency signals, referred to in the literature as the time/frequency Zak transforms. We denote them by: Z _(t) :H _(z) →L ₂(t∈

),  (2.14) Z _(f) :H _(z) →L ₂(f∈

),  (2.15)

As it turns out, the time/frequency Zak transforms are basically geometric projections along the reciprocal dimensions, see FIG. 2. The formulas of the transforms are as follows: Z _(t)(φ)(t)=∫₀ ^(ν) ^(t) φ(t,ν)dν,  (2.16) Z _(f)(φ)(f)=∫₀ ^(τ) ^(r) exp(−j2πfτ)φ(τ,f)dτ,  (2.17)

for every φ∈H_(z). In words, the time Zak transform is integration along the Doppler dimension (taking the DC component) for every point of time. Reciprocally, the frequency Zak transform is Fourier transform along the delay dimension. The formulas of the inverse transforms are as follows:

$\begin{matrix} {{{{Z_{t}^{- 1}(\varphi)}\left( {\tau,v} \right)} = {\sum\limits_{n \in Z}\;{{\exp\left( {{- j}\; 2\;\pi\; v\;\tau_{r}n} \right)}{\varphi\left( {\tau + {n\;\tau_{r}}} \right)}}}},} & (2.18) \\ {{{{Z_{f}^{- 1}(\varphi)}\left( {\tau,v} \right)} = {\sum\limits_{n \in Z}\;{{\exp\left( {j\; 2\;{\pi\tau}\;\left( {{v_{r}n} + v} \right)} \right)}{\varphi\left( {v + {n\; v_{r}}} \right)}}}},} & (2.19) \end{matrix}$

for every φ∈L₂(

). From this point on we will focus only on the time Zak transform and we will denote it by Z=Z_(t). As an intertwining transform Z interchanges between the two Heisenberg operators π_(z)(ν, z) and π_(t)(ν, z), i.e.: Z·π _(z)(ν)=π_(t)(ν)·Z,  (2.20)

for every ν∈V. From the point of view of representation theory the characteristic property of the Zak transform is the interchanging equation (2.20).

A2.5 Standard Zak signal Our goal is to describe the Zak representation of the window function:

$\begin{matrix} {{p(t)} = \left\{ {\begin{matrix} 1 & {0 \leq t < \tau_{r}} \\ 0 & {otherwise} \end{matrix},} \right.} & (2.21) \end{matrix}$

This function is typically used as the generator waveform in multi-carrier modulations (without CP). A direct application of formula (2.18) reveals that P=Z⁻¹(p) is given by:

$\begin{matrix} {{{P\left( {\tau,v} \right)} = {\sum\limits_{n \in Z}\;{{\psi\left( {v\; n\;\tau_{r}} \right)}{p\left( {\tau - {n\;\tau_{r}}} \right)}}}},} & (2.22) \end{matrix}$

One can show that P(aτ_(r),bν_(r))=1 for every a, b∈[0,1), which means that it is of constant modulo 1 with phase given by a regular step function along τ with constant step given by the Doppler coordinate ν. Note the discontinuity of P as it jumps in phase at every integer point along delay. This phase discontinuity is the Zak domain manifestation of the discontinuity of the rectangular window p at the boundaries.

A3. OTFS

The OTFS transceiver structure depends on the choice of the following parameters: a critically sampled lattice Λ=

(τ_(r),0)⊕

(0,ν_(r)), a filter function w∈C(V) and an information grid specified by N, M∈N. We assume that the filter function factorizes as w(τ,ν)=w_(t)(τ)w_(ν)(ν) where the delay and Doppler factors are square root Nyquist with respect to Δτ=τ_(r)/N and Δν=ν_(r)/M respectively. We encode the information bits as a periodic 2D sequence of QAM symbols x=x[nΔτ,mΔν] with periods (N,M). Multiplying x by the standard Zak signal P we obtain a Zak signal x P. A concrete way to think of x P is as the unique quasi periodic extension of the finite sequence x[nΔτ,mΔν] where n=0, . . . , N−1 and m=0, . . . , M−1. We define the modulated transmit waveform as:

$\begin{matrix} \begin{matrix} {{M(x)} = {Z\left( {{\Pi_{\mathcal{z}}(w)}*_{\sigma}{x \cdot P}} \right)}} \\ {{= {Z\left( {w*_{\sigma}{x \cdot P}} \right)}},} \end{matrix} & (3.1) \end{matrix}$

To summarize: the modulation rule proceeds in three steps. In the first step the information block x is quasi-periodized thus transformed into a discrete Zak signal. In the second step, the bandwidth and duration of the signal are shaped through a 2D filtering procedure defined by twisted convolution with the pulse w. In the third step, the filtered signal is transformed to the time domain through application of the Zak transform. To better understand the structure of the transmit waveform we apply few simple algebraic manipulations to (3.1). First, we note that, being an intertwiner (Formula (2.20)), the Zak transform obeys the relation: Z(π_(z)(w)*_(σ) x·p)=Π_(t)(w)

Z(x·P),  (3.2)

Second, we note that the factorization w(τ,ν)=w_(r)(τ)w_(ν)(ν) can be expressed as twisted convolution w=w_(τ)*_(σ) w_(ν). Hence, we can write:

$\begin{matrix} \begin{matrix} {{{\Pi_{t}(w)} \vartriangleright {Z\left( {x \cdot P} \right)}} = {{\Pi_{t}\left( {w_{\tau}*_{\sigma}w_{v}} \right)} \vartriangleright {Z\left( {x \cdot P} \right)}}} \\ {= {{\Pi_{t}\left( w_{\tau} \right)} \vartriangleright \left\{ {{\Pi_{t}\left( w_{v} \right)} \vartriangleright {Z\left( {x \cdot P} \right)}} \right\}}} \\ {{= {w_{\tau}*\left\{ {W_{t} \cdot {Z\left( {x \cdot P} \right)}} \right\}}},} \end{matrix} & (3.3) \end{matrix}$

where W_(t)=FT⁻¹(w_(ν)) and * stands for linear convolution in time. We refer to the waveform Z(x·P) as the bare OTFS waveform. We see from Formula (3.3) that the transmit waveform is obtained from the bare waveform through windowing in time followed by convolution with a pulse. This cascade of operations is the time representation of 2D filtering in the Zak domain. It is beneficial to study the structure of the bare OTFS waveform in the case x is supported on a single grid point (aka consists of a single QAM symbol), i.e., x=δ(nΔτ,mΔν). In this case, one can show that the bare waveform takes the form:

$\begin{matrix} {{{Z\left( {x \cdot P} \right)} = {\sum\limits_{K}\;{{\exp\left( {j\; 2\;\pi\;{m\left( {K + {n\text{/}N}} \right)}\text{/}M} \right)}{\delta\left( {{K\;\tau_{r}} + {n\;{\Delta\tau}}} \right)}}}},} & (3.4) \end{matrix}$

In words, the bare waveform is a shifted and phase modulated infinite delta pulse train of pulse rate ν_(r)=τ_(r) ⁻¹ where the shift is determined by the delay parameter n and the modulation is determined by the Doppler parameter m. Bare and filtered OTFS waveforms corresponding to a single QAM symbol are depicted in FIG. 6 and FIG. 7 respectively. We next proceed to describe the de-modulation mapping. Given a received waveform φ_(rx), its de-modulated image y=D(φ_(rx)) is defined through the rule: D(φ_(rx))=w ^(★)*_(σ) Z ⁻¹(φ_(rx)),  (3.5)

where w^(★) is the matched filter given by w^(★)(ν)=exp(−j2πβ(ν,ν))w(−ν). We often incorporate an additional step of sampling y at (nΔτ,mΔν) for n=0, . . . , N−1 and m=0, . . . , M−1.

A3.1 OTFS channel model The OTFS channel model is the explicit relation between the input variable x and the output variable y in the presence of a channel H. We assume the channel transformation is defined as H=Π_(t) (h) where h=h(τ,ν) is the delay Doppler impulse response. This means that given a transmit waveform φ_(tx), the received waveform φ_(rx)=H (φ_(tx)) is given by:

$\begin{matrix} {{{\varphi_{rx}(t)} = {\int\limits_{\tau,v}{{h\left( {\tau,v} \right)}{\exp\left( {j\; 2\;\pi\;{v\left( {t - \tau} \right)}} \right)}{\varphi_{tx}(t)}d\;\tau\; d\; v}}},} & (3.6) \end{matrix}$

If we take the transmit waveform to be φ_(tx)=M(x) then direct computation reveals that:

$\begin{matrix} \begin{matrix} {y = {D \diamond H \diamond {M(x)}}} \\ {= {w^{\bigstar}*_{\sigma}{Z^{- 1}\left( {{\Pi_{t}(h)} \vartriangleright {Z\left( {w*_{\sigma}{x \cdot P}} \right)}} \right)}}} \\ {= {w^{\bigstar}*_{\sigma}h*_{\sigma}{Z^{- 1} \diamond {Z\left( {w*_{\sigma}{x \cdot P}} \right)}}}} \\ {{= {w^{\bigstar}*_{\sigma}h*_{\sigma}w*_{\sigma}{x \cdot P}}},} \end{matrix} & (3.7) \end{matrix}$

If we denote h_(w)=w^(★*) _(σ)h*_(σ)w then we can write the input-output relation in the form: y=h _(w)*_(σ) x·P,  (3.8)

The delay Doppler impulse h_(w) represents the filtered channel that interacts with the QAM symbols when those are modulated and de-modulated through the OTFS transceiver cycle. One can show that under some mild assumptions h_(w) is well approximated by h*w⁽²⁾ where * stands for linear convolution and w⁽²⁾=w^(★)*w is the linear auto-correlation function. In case the channel is trivial, that is h=δ(0,0), we get that k, =w^(★)*_(σ) w˜w⁽²⁾, thus after sampling we get (an approximate) perfect reconstruction relation: y[nΔτ,mΔν]˜x[nΔτ,mΔν],  (3.9)

for every n=0, . . . , N−1 and m=0, . . . , M−1.

A4. Symplectic Fourier Duality

In this section we describe a variant of the OTFS modulation that can be expressed by means of symplectic Fourier duality as a pre-processing step over critically sampled MC modulation. We refer to this variant as OTFS-MC. For the sake of concreteness, we develop explicit formulas only for the case of OFDM without a CP.

A4.1 Symplectic Fourier transform We denote by L₂ (V) the Hilbert space of square integrable functions on the vector space V. For every ν∈V we define the symplectic exponential (wave function) parametrized by ν as the function ψ_(ν):V→

given by: ψ_(ν)(u)=exp(j2πω(ν,u)),  (4.1)

for every u∈V. Concretely, if ν=(τ,ν) and u=(τ′,ν′) then ψ_(ν)(u)=exp(j2π(ντ′−τν′)). Using symplectic exponents we define the symplectic Fourier transform as the unitary transformation SF:L₂(V)→L₂(V) given by:

$\begin{matrix} \begin{matrix} {{{{SF}(g)}(v)} = {\int\limits_{\upsilon^{\prime}}{\overset{\_}{\psi_{\upsilon}\left( v^{\prime} \right)}{g\left( v^{\prime} \right)}d\; v^{\prime}}}} \\ {{= {\int\limits_{\upsilon^{\prime}}{{\exp\left( {{- j}\; 2\;\pi\;{\omega\left( {v,v^{\prime}} \right)}} \right)}{g\left( v^{\prime} \right)}d\; v^{\prime}}}},} \end{matrix} & (4.2) \end{matrix}$

The symplectic Fourier transform satisfies various interesting properties (much in analogy with the standard Euclidean Fourier transform). The symplectic Fourier transform converts between linear convolution and multiplication of functions, that is: SF(g ₁ *g ₂)=SF(g ₁)·SF(g ₂),  (4.3)

for every g₁,g₂∈L₂(V). Given a lattice Λ⊂V, the symplectic Fourier transform maps sampled functions on A to periodic function with respect to the symplectic reciprocal lattice Λ^(⊥). That is, if g is sampled and G=SF(g) then G(ν+λ^(⊥))=G(ν) for every ν∈V and λ^(⊥)∈Λ^(⊥). This relation takes a simpler form in case A is critically sampled since Λ^(⊥)=Λ. Finally, unlike its Euclidean counterpart, the symplectic Fourier transform is equal to its inverse, that is SF=SF⁻¹.

A4.2 OTFS-MC The main point of departure is the definition of the filtering pulse w and the way it applies to the QAM symbols. To define the MC filtering pulse we consider sampled window function W: Λ→

on the lattice Λ=

τ_(r)⊕

ν_(r). We define w to be the symplectic Fourier dual to W: w=SF(W),  (4.4)

By definition, w is a periodic function on V satisfying w w(ν+λ)=w(ν) for every ν∈V and λ∈A. Typically, W is taken to be a square window with 0/1 values spanning over a certain bandwidth B=M·ν_(r) and duration T=N·τ_(r). In such a case, w will turn to be a Dirichlet sinc function that is Nyquist with respect to the grid Λ_(N,M)=

Δτ⊕

Δν, where: Δτ=τ_(r) /N,  (4.5) Δν=ν_(r) /M,  (4.6)

More sophisticated windows designs can include tapering along the boundaries and also include pseudo-random scrambling phase values. As before, the bits are encoded as a 2D periodic sequence of QAM symbols x=x[nΔτ,mΔν] with period (N, M). The transmit waveform is defined through the rule: M _(MC)(x)=Z((w*x)·P),  (4.7)

In words, the OTFS-MC modulation proceeds in three steps. First step, the periodic sequence is filtered by means of periodic convolution with the periodic pulse w. Second step, the filtered function is converted to a Zak signal by multiplication with the Zak signal P. Third step, the Zak signal is converted into the physical time domain by means of the Zak transform. We stress the differences from Formula (3.1) where the sequence is first multiplied by P and then filtered by twisted convolution with a non-periodic pulse. The point is that unlike (3.1), Formula (4.7) is related through symplectic Fourier duality to MC modulation. To see this, we first note that w*x=SF(W·X) where X=SF(x). This means that we can write:

$\begin{matrix} \begin{matrix} {{\left( {w*x} \right) \cdot P} = {\sum\limits_{\lambda \in \Lambda}\;{{W(\lambda)}{X(\lambda)}{\psi_{\lambda} \cdot P}}}} \\ {{= {{\sum\limits_{\lambda \in \Lambda}\;{{W(\lambda)}{X(\lambda)}{\pi_{z}(\lambda)}}} \vartriangleright P}},} \end{matrix} & (4.8) \end{matrix}$

where the first equality is by definition of the Symplectic Fourier transform and the second equality is by Formula (2.12). We denote Xw=W·X Having established this relation we can develop (4.7) into the form:

$\begin{matrix} \begin{matrix} {{M_{MC}(x)} = {Z\left( {{\sum\limits_{\lambda \in \Lambda}\;{{X_{W}(\lambda)}{\pi_{z}(\lambda)}}} \vartriangleright P} \right)}} \\ {= {\sum\limits_{\lambda \in \Lambda}\;{{X_{W}(\lambda)}{Z\left( {{\pi_{z}(\lambda)} \vartriangleright P} \right)}}}} \\ {= {{\sum\limits_{\lambda \in \Lambda}\;{{X_{W}(\lambda)}{\pi_{t}(\lambda)}}} \vartriangleright {Z\;(P)}}} \\ {{= {{\sum\limits_{\lambda \in \Lambda}\;{{X_{W}(\lambda)}{\pi_{t}(\lambda)}}} \vartriangleright p}},} \end{matrix} & (4.9) \end{matrix}$

where the third equality is the intertwining property of the Zak transform and the forth equality is by definition p=Z(P). In case of OFDM without CP, the pulse p is given by the square window along the interval [0, τ_(r)]. Consequently, the last expression in (4.9) can be written explicitly as:

$\begin{matrix} \begin{matrix} {{M_{MC}(x)} = {{\sum\limits_{k,l}\;{{X_{W}\left( {{k\;\tau_{r}},{l\; v_{r}}} \right)}{\pi_{t}\left( {{k\;\tau_{r}},{l\; v_{r}}} \right)}}} \vartriangleright 1_{\lbrack{0,\tau_{r}}\rbrack}}} \\ {= {{\sum\limits_{k,l}\;{{X_{W}\left( {{k\;\tau_{r}},{l\; v_{r}}} \right)}L_{k\;\tau_{r}}M_{l\; v_{r}}}} \vartriangleright 1_{\lbrack{0,\tau_{r}}\rbrack}}} \\ {{= {\sum\limits_{k,l}\;{{X_{W}\left( {{k\;\tau_{r}},{l\; v_{r}}} \right)}{\exp\left( {j\; 2\;\pi\; l\;{v_{r}\left( {t - {k\;\tau_{r}}} \right)}} \right)}\; 1_{\lbrack{{k\;\tau_{r}},{{({k + 1})}\tau_{r}}}\rbrack}}}},} \end{matrix} & (4.10) \end{matrix}$

The reader can recognize the last expression of (4.10) as MC modulation of the (windowed) sequence of Fourier coefficients Xw. It is interesting to compare the transmit waveforms of OTFS and OTFS-MC corresponding to single QAM symbols. The two structures are depicted in FIG. 8. The main structural difference is the presence of discontinuities at the grid points

τ_(r) in the case of OTFS-MC.

B0. Introduction to OTFS Transceiver Operations from Realization Theory Perspective

In the subsequent sections, we introduce yet another mathematical interpretation of the OTFS transceiver from the point of view of realization theory. In a nutshell, in this approach one considers the signal space of waveforms as a representation space of the Heisenberg group or equivalently as a Hilbert space equipped with collection of Heisenberg operators, each associated with a different point in the delay Doppler plane. This representation space admits multitude of realizations. The two standard ones are the time and frequency realizations and they are related through the one-dimensional Fourier transform. In communication theory the TDMA transceiver structure is naturally adapted to the time realization as QAM symbols are multiplexed along the time coordinate while the OFDM transceiver structure is naturally adapted to the frequency realization as the QAM symbols are multiplexed along the frequency coordinate. The main observation is that, there is a canonical realization lying in between the time and frequency realizations, called the Zak realization. Interestingly, waveforms in Zak realization are represented as functions on a two-dimensional delay Doppler domain satisfying certain quasi-periodicity condition. The main message of this note is that the Zak realization is naturally adapted to the OTFS transceiver. Viewing the OTFS transceiver from this perspective extenuates its novel and independent standing among the other existing transceiver structures. For convenience, we summarize in the following table the main formulas presented in this note:

(0.1) QP φ(v + λ) = ψ (β(v,λ))π_(ϵ) (λ)⁻¹

 φ(v) Z-Heis π^(ϵ) (v₀)

 φ(v) = ψ (−β(v₀v₀))ψ(β(v₀,v))φ(v − v₀) Z-Heis (lattice) π^(ϵ) (λ, ϵ(λ))

 φ (v) = ψ(ω(λ,v))φ(v) Zak to time Z_(time,ϵ) (φ)(t) = ∫₀ ^(vr) φ(t,v)dv time to Zak Z_(ϵ,time) (φ)(t,v) = Σ_(n)ψ(−vτ_(r)n)ƒ(τ + nτ_(r)) Zak to freq Z_(freq,ϵ) (φ)(ƒ) = ∫₀ ^(τr) ψ(−ƒτ)φ(τ,ƒ)dτ freq to Zak Z_(ϵ,freq) (φ)(τ, v) = ψ(τv)Σ_(n)ψ(τv_(r)n)φ(ƒ + nv_(r)) N-Zak to Zak Z_(ϵ,ϵ′) (φ)(τ, v) = φ₀(τ, v) Zak to N-Zak Z_(ϵ′,ϵ) (φ)_(i) (τ,v) = ψ(−v · i/N)φ(τ + i/N,v) Z-std window P_(std) (τ,v) = Σ_(n)ψ(vτ_(r)n)1_([n,n + 1]) (τ/τ_(r))

where the Q abbreviate Quasi and Z abbreviate Zak.

B1. Mathematical preliminaries

B1.1 The Delay Doppler plane Let V=

² be the delay Doppler plane equipped with the standard symplectic bilinear form ω:V×V→

given by: ω(ν₁,ν₂)=ν₁τ₂−τ₁ν₂,  (1.1)

for every ν₁=(τ₁,ν₁) and ν₂=(τ₂,ν₂). Another way to express w is to arrange the vectors ν₁ and ν₂ as the columns of a 2×2 matrix. The symplectic pairing ω(ν₁,ν₂) is equal the additive inverse of the determinant of this matrix, i.e.:

${{\omega\left( {v_{1},v_{2}} \right)} = {- {\det\begin{bmatrix} | & | \\ v_{1} & v_{2} \\ | & | \end{bmatrix}}}},$

We note that the symplectic form is anti-symmetric, i.e., ω(ν₁,ν₂)=−ω(ν₂,ν₁) thus, in particular ω(ν,ν)=0 for every ν∈V. In addition, we consider the polarization form β:V×V→

given by: β(ν₁,ν₂)=ν₁τ₂,  (1.2)

for every ν₁=(τ₁,ν₁) and ν₂=(τ₂,ν₂). We have that: β(ν₁,ν₂)−β(ν₂,ν₁)=ω(ν₀,ν₂),  (1.3)

The form β should be thought of as “half” of the symplectic form. Finally, we denote by ψ(z)=exp(2πiz) is the standard one-dimensional Fourier exponent.

B1.2 Delay Doppler Lattices Refer to Section A2.1 above.

B1.3 The Heisenberg group The polarization form β:V×V→

gives rise to a two-step unipotent group called the Heisenberg group. As a set, the Heisenberg group is realized as Heis=V×S¹ where the multiplication rule is given by: (ν₁ ,z ₁)·(ν₂ ,z ₂)=(ν₁+ν₂,ψ(β(ν₁,ν₂))z ₁ z ₂),  (1.11)

One can verify that indeed the rule (1.11) induces a group structure, i.e., it is associative, the element (0, 1) acts as unit and the inverse of (ν,z) is (−ν,ψ(β(ν,ν))z⁻¹). We note that the Heisenberg group is not commutative, i.e., {tilde over (ν)}₁·{tilde over (ν)}₂ is not necessarily equal to {tilde over (ν)}₂·{tilde over (ν)}₁. The center of the group consists of all elements of the form (0,z), z∈S¹. The multiplication rule gives rise to a group convolution operation between functions:

$\begin{matrix} {{{{f_{1}\mspace{11mu}\mspace{14mu}{f_{2}\left( \overset{\sim}{v} \right)}} = {\int_{{\overset{\sim}{v_{1}} \cdot {\overset{\sim}{v}}_{2}} = \overset{\sim}{\upsilon}}{{f_{1}\left( {\overset{\sim}{v}}_{1} \right)}{f_{2}\left( {\overset{\sim}{v}}_{2} \right)}}}},}\ } & (1.12) \end{matrix}$

for every pair of functions ƒ₁, ƒ₂∈

(Heis). We refer to the convolution operation

as Heisenberg convolution or twisted convolution.

The Heisenberg group admits multitude of finite subquotient groups. Each such group is associated with a choice of a pair (Λ,ϵ) where ΛV is an under-sampled lattice and ϵ:Λ→S¹ is a map satisfying the following condition: ϵ(λ₁+λ₂)=ϵ(λ₁)ϵ(λ₂)ψ(β(λ₁,λ₂)),  (1.13)

Using ϵ we define a section map {circumflex over (ϵ)}:Λ→Heis given by {circumflex over (ϵ)}(λ)=(λ,ϵ(λ)). One can verify that (1.13) implies that {circumflex over (ϵ)} is a group homomorphism, that is {circumflex over (ϵ)}(λ₁+λ₂)={circumflex over (ϵ)}(λ₁)·{circumflex over (ϵ)}(λ₂). To summarize, the map ϵ defines a sectional homomorphic embedding of A as a subgroup of the Heisenberg group. We refer to ϵ as a Heisenberg character and to the pair (λ,ϵ) as a Heisenberg lattice. A simple example is when the lattice Λ is rectangular, i.e., Λ=Λ_(r). In this situation β|Λ=0 thus we can take ϵ=1, corresponding to the trivial embedding {circumflex over (ϵ)}(λ)=(λ,1). A more complicated example is the hexagonal lattice Λ=Λ_(hex) equipped with ϵ_(hex):Λ_(hex)→S¹, given by: ϵ_(hex)(ng ₁ +mg ₂)=ψ(m ²/4),  (1.14)

for every n, m∈

. An Heisenberg lattice defines a commutative subgroup Λ_(ϵ)⊂Heis consisting of all elements of the form (λ,ϵ(λ)), λ∈Λ. The centralizer subgroup of Im{circumflex over (ϵ)} is the subgroup Λ^(⊥)×S¹. We define the finite subquotient group: Heis(Λ,ϵ)=Λ^(⊥) ×S ¹/Im{circumflex over (ϵ)},  (1.15)

The group Heis (Λ,ϵ) is a central extension of the finite commutative group Λ^(⊥)/Λ by the unit circle S¹, that is, it fits in the following exact sequence: S ^(1′)Heis(Λ,ϵ)

Λ^(⊥)/Λ,  (1.16)

We refer to Heis (Λ,ϵ) as the finite Heisenberg group associated with the Heisenberg lattice (Λ,ϵ). The finite Heisenberg group takes a more concrete from in the rectangular case. Specifically, when Λ=Λ_(r) and ϵ=1, we have Heis(Λ,1)=Λ^(⊥)/Λ×S¹=

/N×

/N×S¹ with multiplication rule given by: (k ₁ ,l ₁ ,z ₁)·(k ₂ ,l ₂ ,z ₂)=(k ₁ +k ₂ ,l ₁ +l ₂,ψ(l ₁ k ₂ /N)z ₁ z ₂),  (1.17)

B1.4 The Heisenberg representation The representation theory of the Heisenberg group is relatively simple. In a nutshell, fixing the action of the center, there is a unique (up-to isomorphism) irreducible representation. This uniqueness is referred to as the Stone-von Neumann property. The precise statement is summarized in Section A1.3: The fact that π is a representation, aka multiplicative, translates to the fact that Π interchanges between group convolution of functions and composition of linear transformations, i.e., Π(ƒ₁

ƒ₂)=Π(ƒ₁)·Π(ƒ₂). Since π(0, z)=zId_(H), by Fourier theory, its enough to consider only functions ƒ that satisfy the condition ƒ(ν, z)=z⁻¹ƒ(ν,1). Identifying such functions with their restriction to V=V×{1} we can write the group convolution in the form:

$\begin{matrix} {{{f_{1}\mspace{11mu}\mspace{14mu}{f_{2}(v)}} = {\int_{{v_{1} + v_{2}} = v}\ {{\psi\left( {\beta\left( {v_{1},v_{2}} \right)} \right)}{f_{1}\left( v_{1} \right)}{f_{2}\left( v_{2} \right)}}}},} & (1.19) \end{matrix}$

Interestingly, the representation π, although is unique, admits multitude of realizations. Particularly well known are the time and frequency realizations which are omnipresent in signal processing. We consider the Hilbert space of complex valued functions on the real line H=

(

). To describe them, we introduce two basic unitary operations on such functions—one called delay and the other modulation, defined as follows: Delay: L _(x)(φ)(y)=φ(y−x),  (1.20) Modulation: M _(x)(φ)(y)=ψ(xy)φ(y),  (1.21)

for any value of the parameter x∈

and every function φ∈H. Given a point ν=(τ,ν) we define the time realization of the Heisenberg representation by: π^(time)(ν,z)

>z·Lτ·M _(ν)(φ)  (1.22)

where we use the notation

to designate the application of an operator on a vector. It is accustom in this context to denote the basic coordinate function by t (time). Under this convention, the right hand side of (1.22) takes the explicit form zψ(ν(t−τ))φ(t−τ). Reciprocally, we define the frequency realization of the Heisenberg representation by: π^(freq)(ν,z)

φ=z·M _(−τ) ·L _(ν)(φ),  (1.23)

In this context, it is accustom to denote the basic coordinate function by ƒ (frequency). Under this convention, the right hand side of (1.23) takes the explicit form zψ(−τƒ)φ(ƒ−ν). By Theorem 1.1, the time and frequency realizations are isomorphic. The isomorphism is given by the Fourier transform: FT(φ)(ƒ)=∫_(t) exp(−2πift)φ(t)dt,  (1.24)

for every φ∈H. As an intertwining transform FT interchanges between the two Heisenberg operators π_(t)(ν,z) and π_(f)(ν,z), i.e.: FT·π ^(time)(ν,z)=π^(freq)(ν,z)·FT,  (1.25)

for every (ν,z). From the point of view of representation theory the characteristic property of the Fourier transform is the interchanging equation (1.25). Finally, we note that from communication theory perspective, the time domain realization is adapted to modulation techniques where QAM symbols are arranged along a regular lattice of the time domain. Reciprocally, the frequency realization is adapted to modulation techniques (line OFDM) where QAM symbols are arranged along a regular lattice on the frequency domain. We will see in the sequel that there exists other, more exotic, realizations of the signal space which give rise to a family of completely new modulation techniques which we call ZDMA.

The finite Heisenberg representation. It is nice to observe that the theory of the Heisenberg group carry over word for word to the finite set-up. In particular, given an Heisenberg lattice (Λ,ϵ), the associated finite Heisenberg group Heis (Λ,ϵ) admits a unique up to isomorphism irreducible representation after fixing the action of the center. This is summarized in the following theorem.

Theorem 1.2 (Finite dimensional Stone-von Neumann theorem). There is a unique (up to isomorphism) irreducible unitary representation π_(ϵ): Heis(Λ,ϵ)→U(H) such that π_(ϵ)(0,z)=zId_(H). Moreover π_(ϵ) is finite dimensional with dim H=N where N²=#Λ^(⊥)/Λ.

For the sake of simplicity, we focus our attention on the particular case where Λ=

(τ_(r),0)⊕

(0,ν_(r)) is rectangular and ϵ=1 and proceed to describe the finite dimensional counterparts of the time and frequency realizations of π_(ϵ). To this end, we consider the finite dimensional Hilbert space H_(N)=

(

/N) of complex valued functions on the ring

/N, aka—the finite line. Vectors in H_(N) can be viewed of as uniformly sampled functions on the unit circle. As in the continuous case, we introduce the operations of (cyclic) delay and modulation: L _(n)(φ)(m)=φ(m−n)  (1.26) M _(n)(φ)(m)=ψ(nm/N)φ(m),  (1.27)

for every φ∈H_(N) and n∈

/N noting that the operation m−n is carried in the cyclic ring

/N. Given a point (k/ν_(r),l/τ_(r))∈Λ^(⊥) we define the finite time realization by: π_(ϵ) ^(time)(k/ν _(r) ,l/τ _(r) ,z)t)

φ=z·L _(k) ·M ₁(φ)  (1.28)

Denoting the basic coordinate function by n we can write the right hand side of (1.28) in the explicit form zψ(l(n−k)/N)φ(n−k). Reciprocally, we define the finite frequency realization by: π_(ϵ) ^(freq)(k/ν _(r) ,l/τ _(r) ,z)

φ=z·M _(−k) ·L _(l)(φ),  (1.29)

Denoting the basic coordinate function by m the right hand side of (1.29) can be written in the explicit form zψ(−km/N)φ(m−l). By Theorem 1.2, the discrete time and frequency realizations are isomorphic and the isomorphism is realized by the finite Fourier transform:

$\begin{matrix} {{{{{FFT}(\varphi)}(m)} = {\sum\limits_{n = 0}^{N - 1}{\exp\mspace{11mu}\left( {{- 2}\pi\;{imn}\text{/}N} \right)\mspace{11mu}\varphi\mspace{11mu}(n)}}},} & (1.30) \end{matrix}$

As an intertwining transform the FFT interchanges between the two Heisenberg operators π_(ϵ) ^(time)(ν, z) and π_(ϵ) ^(freq)(ν, z), i.e.: FFT·π _(ϵ) ^(time)(ν,z)=π_(ϵ) ^(freq)(ν,z)·FFT,  (1.31)

for every (ν,z) ∈ Heis(Λ,ϵ).

B2. The Zak realization

B2.1 Zak waveforms See previous discussion in A2.2. In this section we describe a family of realizations of the Heisenberg representation that simultaneously combine attributes of both time and frequency. These are known in the literature as Zak type or lattice type realizations. A particular Zak realization is parametrized by a choice of an Heisenberg lattice (Λ, ϵ) where A is critically sampled. A Zak waveform is a function φ: V→

that satisfies the following quasi periodicity condition: (2.1)φ(ν+λ)=ϵ(λ)ψ(β(ν,λ))φ(ν)

There is an alternative formulation of condition (2.1) that is better suited when considering generalizations. The basic observation is that the map E defines a one dimensional representation π_(ϵ):Λ×S¹→U(

) satisfying the extra condition that π_(ϵ)(0,z)=z. This representation is given by π_(ϵ)(λ,z)=ϵ(λ)⁻¹ z. Indeed, verify that:

$\begin{matrix} \begin{matrix} {{{\pi_{\epsilon}\left( {\lambda_{1},z_{1}} \right)} \cdot {\pi_{\epsilon}\left( {\lambda_{2},z_{2}} \right)}} = {\epsilon\mspace{11mu}\left( \lambda_{1} \right)^{- 1}\epsilon\mspace{11mu}\left( \lambda_{2} \right)^{- 1}z_{1}z_{2}}} \\ {= {{\epsilon\left( {\lambda_{1} + \lambda_{2}} \right)}^{- 1}\mspace{11mu}\psi\mspace{11mu}\left( {\beta\left( {\lambda_{1},\lambda_{2}} \right)} \right)\mspace{11mu} z_{1}z_{2}}} \\ {{= {\pi_{\epsilon}\left( {\left( {\lambda_{1},z_{1}} \right) \cdot \left( {\lambda_{2},z_{2}} \right)} \right)}},} \end{matrix} & (2.2) \end{matrix}$

In addition, we have that π_(ϵ)(λ,ϵ(λ))=1 implying the relation Im{circumflex over (ϵ)}⊂ker π_(ϵ). Hence π_(ϵ), is in fact a representation of the finite Heisenberg group Heis (Λ,ϵ)=Λ×S¹/Im{circumflex over (ϵ)}. Using the representation π_(ϵ), we can express (2.1) in the form: φ(ν+λ)=ψ(β(ν,λ)){π_(ϵ)(λ)⁻¹

φ(ν)}  (2.3)

We denote the Hilbert space of Zak waveforms by H (V,π_(ϵ)) or sometimes for short by 15 H_(ϵ). For example, in the rectangular situation where Λ=Λ_(r) and ϵ=1, condition (2.1) takes the concrete form φ(τ+kτ_(r),ν+lν_(r))=ψ(νkτ_(r))φ(τ,ν), that is, φ is periodic function along the Doppler dimension (with period ν,) and quasi-periodic function along the delay dimension. Next, we describe the action of the Heisenberg group on the Hilbert space of Zak waveforms. Given a Zak waveform φ∈H_(ϵ), and an element (u, z) ∈ Heis, the action of the element on the waveform is given by: {π^(ϵ)(u,z)

φ}(ν)=z·ψ(β(u,ν−u))φ(ν−u),  (2.4) In addition, given a lattice point λ∈Λ, the action of the element {circumflex over (ϵ)}(λ)=(λ,ϵ(λ)) takes the simple form:

$\begin{matrix} \begin{matrix} {{\left\{ {{\pi^{\epsilon}\left( {\lambda_{1}{\epsilon(\lambda)}} \right)} \vartriangleright \varphi} \right\}(v)} = {{\epsilon(\lambda)}{\psi\left( {\beta\left( {\lambda,{v - \lambda}} \right)} \right)}{\varphi\left( {v - \lambda} \right)}}} \\ {= {{\epsilon(\lambda)}{\epsilon\left( {- \lambda} \right)}{\psi\left( {\beta\left( {\lambda,{- \lambda}} \right)} \right)}{\psi\left( {\omega\left( {\lambda,v} \right)} \right)}{\varphi(\upsilon)}}} \\ {{= {{\psi\left( {\omega\left( {\lambda,v} \right)} \right)}{\varphi(v)}}},} \end{matrix} & (2.5) \end{matrix}$

where in the first equality we use (2.4), in the second equality we use (2.1) and the polarization equation (1.3) and in the third equality we use (1.13). To conclude, we see that π^(ϵ)({circumflex over (ϵ)}(λ)) is given by multiplication with the symplectic Fourier exponent associated with the point λ. As usual, the representation gives rise to an extended action by functions on V. Given a function h∈

(V), its action on a Zak waveform:

$\begin{matrix} \begin{matrix} {{\left\{ {{\pi^{\epsilon}(h)} \vartriangleright \varphi} \right\}(v)} = {\int_{{u\epsilon}\; v}{{h(u)}\left\{ {{\pi^{\epsilon}(u)} \vartriangleright \varphi} \right\}(v)\;{du}}}} \\ {{= {\int_{{u\epsilon}\; v}{{\psi\left( {\beta\left( {u,{v - u}} \right)} \right)}{h(u)}{\varphi\left( {v - u} \right)}\;{du}}}},} \end{matrix} & (2.6) \end{matrix}$

From the last expression we conclude that Π^(ϵ)(h)

φ=h

φ, namely, the extended action is realized by twisted convolution of the impulse h with the waveform ν.

B2.2 Zak transforms See also section A2.4. By Theorem 1.1, the Zak realization is isomorphic both to the time and frequency realizations. Hence there are intertwining transforms interchanging between the corresponding Heisenberg group actions. These intertwining transforms are usually referred to in the literature as the time/frequency Zak transforms and we denote them by: Z _(time,ϵ) :H _(ϵ) →H _(time)=

(t∈

),  (2.7) Z _(freq,ϵ) :H _(ϵ) →H _(freq)=

(ƒ∈

),  (2.8)

As it turns out, the time/frequency Zak transforms are basically geometric projections along the reciprocal dimensions, see FIG. 2. Formally, this assertion is true only when the maximal rectangular sublattice Λ_(r)=

(τ_(r),0)⊕

(0,ν_(r)) is non-trivial, i.e., when the rectangular parameters τ_(r),ν_(r)<∞. Assuming this condition holds, let N=τ_(r)·ν_(r) denote the index of the rectangular sublattice A_(r) with respect to the full lattice A, i.e., N=[Λ_(r):Λ]. For example, when Λ=Λ_(rec) we have τ_(r)=ν_(r)=1 and N=1. When Λ=Λ_(hex), we have τ_(r)=a and ν_(r)=2/a and consequently N=2. Without loss of generality, we assume that ϵ|Λ_(r)=1.

Granting this assumption, we have the following formulas: Z _(time,ϵ)(φ)(t)=∫₀ ^(ν) ^(r) φ(t,ν)dv,  (2.9) Z _(freq,ϵ)(φ)(ƒ)=∫₀ ^(τ) ^(r) ψ(−ƒτ)φ(τ,ƒ)dτ,  (2.10)

We now proceed describe the intertwining transforms in the opposite direction, which we denote by: Z _(ϵ,time) :H _(time) →H _(ϵ),  (2.11) Z _(ϵ,freq) :H _(freq) →H _(ϵ),  (2.12)

To describe these we need to introduce some terminology. Let π_(r) ^(time) and b^(freq) denote the time and frequency realizations of the Heisenberg representation of the group Heis (Λ_(r),1) b^(time), b^(freq)∈

^(N) are the unique (up to multiplication by scalar) invariant vectors under the action of Λ_(ϵ) through π_(r) ^(time) and π_(r) ^(freq) respectively. The formulas of (2.11) and (2.12) are:

$\begin{matrix} {{{{Z_{\epsilon,{time}}(\varphi)}\left( {\tau,v} \right)} = {\sum\limits_{k = 0}^{N - 1}{\sum\limits_{n \in {\mathbb{Z}}}{{b^{time}\lbrack k\rbrack}{\psi\left( {{- v}\;{\tau_{r}\left( {{k\text{/}N} + n} \right)}} \right)}{\varphi\left( {\tau + {\tau_{r}\left( {{k\text{/}N} + n} \right)}} \right)}}}}},} & (2.13) \\ {{{{Z_{\epsilon,{freq}}(\varphi)}\left( {\tau,v} \right)} = {{\psi\left( {\tau\; v} \right)}{\sum\limits_{k = 0}^{N - 1}{\sum\limits_{n \in {\mathbb{Z}}}{{b^{freq}\lbrack k\rbrack}{\psi\left( {\tau\;{v_{r}\left( {{k\text{/}N} + n} \right)}} \right)}{\varphi\left( {v + {v_{r}\left( {{k\text{/}N} + n} \right)}} \right)}}}}}},} & (2.14) \end{matrix}$

In the rectangular situation where Λ=Λ_(r), and ϵ=1, we have N=1 and b^(time)=b^(freq)=1. Substituting these values in (2.13) and (2.14) we get:

$\begin{matrix} {{{{Z_{\epsilon,{time}}(\varphi)}\left( {\tau,v} \right)} = {\sum\limits_{n \in {\mathbb{Z}}}{{\psi\left( {{- v}\;\tau_{r}n} \right)}{\varphi\left( {\tau + {n\;\tau_{r}}} \right)}}}},} & (2.15) \\ {{{{Z_{\epsilon,{freq}}(\varphi)}\left( {\tau,v} \right)} = {{\psi\left( {\tau\; v} \right)}{\sum\limits_{n \in {\mathbb{Z}}}{{\psi\left( {\tau\; v_{r}n} \right)}{\varphi\left( {v + {nv}_{r}} \right)}}}}},} & (2.16) \end{matrix}$

In addition, in the hexagonal situation where Λ=Λ_(hex) and ϵ=ϵ_(hex), we have N=2, τ_(r)=a, ν_(r)=2a⁻¹ and b^(time)=(1,i), b^(freq)=(1,−i). Substituting these values in (2.11) and (2.12) we get:

$\begin{matrix} {{{{Z_{\epsilon,{time}}(\varphi)}\left( {\tau,v} \right)} = {{\sum\limits_{n \in {\mathbb{Z}}}{{\psi\left( {- {van}} \right)}{\varphi\left( {\tau + {an}} \right)}}} + {i{\sum\limits_{n \in {\mathbb{Z}}}{{\psi\left( {- {{va}\left( {{1\text{/}2} + n} \right)}} \right)}{\varphi\left( {\tau + {a\left( {{1\text{/}2} + n} \right)}} \right)}}}}}},} & (2.17) \\ {{{Z_{\epsilon,{freq}}(\varphi)}\left( {\tau,v} \right)} = {{{\psi\left( {\tau\; v} \right)}{\sum\limits_{n \in {\mathbb{Z}}}{{\psi\left( {2\tau\; a^{- 1}n} \right)}{\varphi\left( {v + {2a^{- 1}n}} \right)}}}} - {i\;{\psi\left( {\tau\; v} \right)}{\sum\limits_{n \in {\mathbb{Z}}}{{\psi\left( {2\tau\;{a^{- 1}\left( {n + {1\text{/}2}} \right)}} \right)}{\varphi\left( {v + {2{a^{- 1}\left( {n + {1\text{/}2}} \right)}}} \right)}}}}}} & (2.18) \end{matrix}$

Furthermore, one can show that Z_(time,ϵ)·Z_(ϵ,freq)∝FT hence the pair of Zak transforms constitute a square root decomposition of the Fourier transform, reinforcing the interpretation of the Zak realization as residing between the time and the frequency (see FIG. 15). As mentioned before, the characteristic property of the Zak transform is that it interchanges between the Heisenberg group actions.

Proposition 2.1. We have: Z _(time,ϵ)(π^(ϵ)(ν,z)

φ)=π^(time)(ν,z)

Z _(time,ϵ)(φ),  (2.19) Z _(freq,ϵ)(π^(ϵ)(ν,z)

φ)=π^(freq)(ν,z)

Z _(freq,ϵ)(φ),  (2.20) for every φ∈H_(ϵ) and (ν, z)∈ Heis

Example 2.2. As an example we consider the rectangular lattice Λ_(r)=

(τ_(r), 0)⊕

(0,1/τ_(r)) and the trivial Heisenberg character ϵ=1. Under these choices, we describe the Zak realization of the window function:

$\begin{matrix} {{p(t)} = \left\{ {\begin{matrix} 1 & {0 \leq t < \tau_{r}} \\ 0 & {otherwise} \end{matrix},} \right.} & (2.21) \end{matrix}$

This function is typically used as the generator filter in multi-carrier modulations (without CP). A direct application of formula (2.15) reveals that P=Z_(ϵ,time)(p) is given by:

$\begin{matrix} {{{P\left( {\tau,v} \right)} = {\sum\limits_{n \in {\mathbb{Z}}}{{\psi\left( {vn\tau}_{r} \right)}{p\left( {\tau - {n\;\tau_{r}}} \right)}}}},} & (2.22) \end{matrix}$

One can show that P (aτ_(r),b/τ_(r))=1 for every a, b∈[0,1), which means that it is of constant modulo 1 with phase given by a regular step function along τ with constant step given by the Doppler coordinate ν. Note the discontinuity of P as it jumps in phase at every integer point along delay. This phase discontinuity is the Zak domain manifestation of the discontinuity of the rectangular window p at the boundaries.

B3. The Generalized Zak Realization

For various computational reasons that arise in the context of channel equalization we need to extend the scope and consider also higher dimensional generalizations of the standard scalar Zak realization. Specifically, a generalized Zak realization is a parametrized by an under-sampled Heisenberg lattice (Λ, ϵ). Given this choice, we fix the following structures:

Let Heis (Λ,ϵ)=Λ^(⊥)×S¹/Λ_(ϵ), be the finite Heisenberg group associated with (Λ,ϵ), see Formula (1.15). Let N²=[Λ:Λ^(⊥)] be the index of Λ inside Λ^(⊥). Finally, let π_(ϵ) be the finite dimensional Heisenberg representation of Heis (Λ,ϵ). At this point we are not interested in any specific realization of the representation π_(ϵ).

A generalized Zak waveform is a vector valued function φ:V→

^(N) that satisfy the following it, quasi-periodicity condition: φ(ν+λ)=ψ(β(ν,λ)){π_(ϵ)(λ)⁻¹

φ(ν)},  (3.1)

for every ν∈V and λ∈Λ^(⊥). Observe that when the lattice A is critically sampled, we have N=1 and condition (3.1) reduces to (2.3). In the rectangular situation where Λ=Λ_(r), ϵ=1 we can take π_(ϵ)=π_(ϵ) ^(time), thus the quasi-periodicity condition (3.1) takes the explicit form: φ(τ+k/ν _(r) ,ν+l/τ _(r))=ψ(νk/ν _(r)){ψ(kl/N)M ⁻¹ L _(−k)

φ(τ,ν)},  (3.2)

where we substitute ν=(τ, ν) and λ=(k/ν_(r),l/τ_(r)). In particular, we see from (3.2) that the nth coordinate of co satisfies the following condition along Doppler: φ_(n)(τ,ν+l/τ _(r))=ψ(−nl/N)φ_(n)(τ,ν),  (3.3)

for every (τ,ν)∈V and l∈

. We denote by H_(ϵ)=H(V,π_(ϵ)) the Hilbert space of generalized Zak waveforms. We now proceed to define the action of Heisenberg group on H_(ϵ). The action formula is similar to (2.4) and is given by: {π^(ϵ)(ν,z)

φ}(ν′)=z·ψ(β(ν,ν′−ν))φ(ν′−ν),  (3.4)

for every φ∈H_(ϵ) and (ν,z)∈ Heis. Similarly, we have {π^(ϵ)(λ,ϵ(λ))

φ}(ν)=ψ(ω(λ,ν))φ(ν), for every λ∈Λ.

3.1 Zak to Zak Intertwining Transforms

The standard and the generalized Zak realizations of the Heisenberg representation are isomorphic in the sense that there exists a non-zero intertwining transform commuting between the corresponding Heisenberg actions. To describe it, we consider the following setup. We fix a critically sampled Heisenberg lattice (Λ,ϵ) and a sub-lattice Λ′⊂Λ of index N. We denote by ϵ′ the restriction of ϵ to the sub-lattice Λ′. Our goal is to describe the intertwining transforms (See FIG. 16): Z _(ϵ,ϵ′) :H _(ϵ′) →H _(ϵ),  (3.5) Z _(ϵ′,ϵ) :H _(ϵ) →H _(ϵ′),  (3.6)

We begin with the description of Z_(ϵ,ϵ′). Let ζ∈

^(N) be the unique (up-to multiplication by scalar) invariant vector under the action of the subgroup V^(ϵ)={circumflex over (ϵ)}(V)⊂ Heis (Λ′,ϵ′) through the representation π_(ϵ′), namely, ζ satisfies the condition: π_(ϵ′)(λ,ϵ)(λ))

ζ=ζ,

for every λ∈Λ. Given a generalized Zak waveform φ∈H_(ϵ′), the transformed waveform Z_(ϵ,ϵ′)(φ) is given by: Z _(ϵ,ϵ′)(φ)(ν)=

ζ,φ(ν)

,  (3.7)

for every ν∈V. In words, the transformed waveform is defined pointwise by taking the inner product with the invariant vector ζ. We proceed with the description of Z_(ϵ′,ϵ). To this end, we define the Hilbert space of sampled Zak waveforms. A sampled Zak waveform is a function ϕ:Λ′^(⊥)→

satisfying the following discrete version of the quasi-periodicity condition (2.3): ϕ(δ+λ)=ψ(β(δ,λ))π_(ϵ)(λ)⁻¹

ϕ(δ),  (3.8)

for every δ∈Λ′^(⊥) and λ∈Λ. We denote the Hilbert space of sampled Zak waveforms by H (Λ′^(⊥),π_(ϵ)). One can show that H(Λ′^(⊥),π_(ϵ)) is a finite dimensional vector space of dimension [Λ:Λ′^(⊥)]=[Λ′:Λ]=N. The Hilbert space of sampled Zak waveforms admits an action of the finite Heisenberg group Heis (Λ′,ϵ′). This action is a discrete version of 2.4) given by: {π_(ϵ)(δ,z)

ϕ}(δ′)=zψ(β(δ,δ′−δ))ϕ(δ′−δ),  (3.9)

for every ϕ∈H (Λ′^(⊥),π_(ϵ)), and points δ,δ′∈Λ′^(⊥). We can now define the intertwining transform Z_(ϵ′,ϵ).

Given a Zak waveform φ∈H_(ϵ) the transformed generalized waveform φ′=Z_(ϵ′,ϵ)(φ) is a function on V taking values in the N dimensional Hilbert space H(Λ′^(⊥),π_(ϵ))≅

^(N), defined by: φ′(ν)(δ)=ψ(−β(ν,δ))φ(ν+δ),  (3.10)

for every ν∈V and δΛ′^(⊥). For the sake of concreteness, it is beneficial to describe in detail the rectangular situation. We consider a rectangular lattice Λ=Λ_(r) with trivial embedding ϵ=1 and the sublattice Λ′=

(τ_(r),0)⊕

(0,Nν_(r)). Evidently, we have [Λ′:Λ]=N. For these particular choices, the structures described above take the following concrete form:

-   -   The finite Heisenberg group associated with (Λ,ϵ) is given by:         Heis(Λ,ϵ)=Λ^(⊥) /Λ×S ¹ ≅S ¹,     -   The finite Heisenberg representation of Heis (Λ,ϵ), is given by:         π_(ϵ)(z)=z,     -   The orthogonal complement lattice of Λ′ is given by:         Λ′^(⊥)=         (τ_(r) /N,0)⊕         (0,ν_(r)),     -   The finite Heisenberg group associated with (Λ′,ϵ′) is given by:         Heis(Λ′,ϵ′)=Λ′^(⊥) /Λ×S ¹ ≅         /N×         /N×S ¹,     -   The finite Heisenberg representation of Heis (Λ′,ϵ′), is given         by π_(ϵ′)=π_(ϵ′) ^(time), where:         π_(ϵ′) ^(time)(kτ _(r) /N,lν _(r) ,z)=zL _(k) M _(l),     -   The invariant vector under π_(ϵ′)(λ,ϵ′(λ))=π_(ϵ′)(λ,ϵ′(λ)), λ∈Λ         is given by:         ζ=δ(0),

Substituting in Formula (3.7), we get:

$\begin{matrix} \begin{matrix} {{{Z_{\epsilon,\epsilon^{\prime}}(\varphi)}\mspace{11mu}(v)} = \left\langle {{\delta(0)},{\varphi(v)}} \right\rangle} \\ {= {\varphi_{0}(v)}} \end{matrix} & (3.11) \end{matrix}$

In words, the conversion from generalized to standard Zak waveforms is “simply” to take the zero coordinate at each point ν∈V. In the opposite direction, given a Zak waveform φ∈H (V,π_(ϵ)) its restriction to the lattice Λ′^(⊥) is periodic with respect to translations by elements of Λ, hence is a function on the quotient group Λ′^(⊥)/Λ=

/N, i.e., a vector in

(

/N). Substituting in Formula (3.10), we get that:

${{{Z_{\epsilon^{\prime},\epsilon}(\varphi)}\mspace{11mu}\left( {\tau,v} \right)} = \begin{bmatrix} {{\psi\left( {{- v}\;\tau_{r}0\text{/}N} \right)}{\varphi\left( {\tau,v} \right)}} \\ {{\psi\left( {{- v}\;\tau_{r}\text{/}N} \right)}{\varphi\left( {{\tau + {\tau_{r}\text{/}N}},v} \right)}} \\ \vdots \\ {{\psi\left( {{- v}\;{\tau_{r}\left( {1 - {1\text{/}N}} \right)}} \right)}{\varphi\left( {{\tau + {\tau_{r}\left( {1 - {1\text{/}N}} \right)}},v} \right)}} \end{bmatrix}},$

for every φ∈H (V,π_(ϵ)) and (τ,ν)∈V.

B4. ZDMA Transceiver Embodiments

In this section we describe the structure of the ZDMA transceiver incorporating the Zak realization formalism. In addition, we describe a weaker version that can be implemented as a preprocessing step over multi-carrier modulation.

B4.1 Transceiver Parameters

The ZDMA transceiver structure is based on the following parameters:

-   -   (1) An Heisenberg critically sampled lattice (Λ,ϵ) giving rise         to the Hilbert space of Zak waveforms H (V,π_(ϵ)).     -   (2) A transmit and receive filter functions w_(tx), w_(rx)∈         (V).     -   (3) A non-degenerate pulse waveform φ∈H (V,π_(ϵ)) satisfying         P(ν)≠0 for every ν∈V.

The transmit function w_(tx) is a function on the delay Doppler plane that plays the role of a two dimensional filter, shaping the transmitted signal to a specific bandwidth and specific time duration. The receive function w_(rx) is principally the matched filter to w_(tx). We will assume, henceforth that it is defined as w_(rx)=w_(tx) ^(★):

$\begin{matrix} \begin{matrix} {{\omega_{rx}(v)} = {\omega_{rx}^{\bigstar}(v)}} \\ {{= {{\psi\left( {- {\beta\left( {v,v} \right)}} \right)}\mspace{14mu}\overset{\_}{\omega_{tx}\left( {- v} \right)}}},} \end{matrix} & (4.1) \end{matrix}$

for every ν∈V. In addition, we assume that the function w=w_(tx) can be decomposed as a twisted/Heisenberg convolution w=w_(z)

w_(ν) where w_(τ) is a one dimensional function supported on the delay axis and w_(ν) is one dimensional function supported on the Doppler axis. One can verify that: w(τ,ν)=w _(τ)(τ)w _(ν)(ν),  (4.2)

for every (τ,ν)∈

²=V. The benefit in considering such decomposable filters is that most of their attributes can be expressed in terms of simple analysis of the two factors. In particular, in this situation the received matched function is given by w_(rx) ^(★)=w_(ν) ^(★)

w_(τ) ^(★) where w_(τ) ^(★) and w_(ν) ^(★) are the respected one dimensional conjugate functions familiar from standard signal processing, i.e.: w _(τ) ^(★)(τ)= w _(τ)(−τ),  (4.3) w _(ν) ^(★)(ν)= w _(ν)(−ν),  (4.4)

for every τ∈R and ν∈

respectively. In typical situation, we require the one dimensional filter w_(τ) to be a square root Nyquist with respect to a bandwidth B>0, i.e., w_(τ) ^(★)*w_(τ)(k/B)=0 for every non-zero integer k, and, reciprocally, we require the one dimensional filter w_(ν) to be square root Nyquist with respect to a duration T>0, i.e., w_(ν) ^(★)*w_(ν)(l/T)≠0 for every non-zero integer l. To proceed further we need to choose a basis of the lattice A: Λ=

g ₁ ⊕

g ₂,  (4.5)

Granting such a choice we can define the pulse P to be the unique quasi-periodic function that satisfy P(ag₁+bg₂) for every 0≤a,b≤1. Note that when the lattice is rectangular and the basis is the standard one, such pulse is described in Example 2.2. Before describing the transceiver structure we need to explain how to encode the information bits. These are encoded into a periodic function x∈

(V) with respect to the lattice λ. In typical situations we assume that x is a sampled Λ—periodic function of the form:

$\begin{matrix} {{x = {\sum\limits_{n,m}{{x\left\lbrack {n,m} \right\rbrack}{\delta\left( {{{ng}_{1}\text{/}N} + {{mg}_{2}\text{/}M}} \right)}}}},} & (4.6) \end{matrix}$

where N, M∈

^(≥1) are fixed parameters defining the density of the delay Doppler information lattice Λ_(N,M)=

g₁/N⊕

g₂/M. In more canonical terms, x is a function on the information lattice Λ_(N,M) that is periodic with respect to the sub-lattice Λ∈Λ_(N,M). The expression for x is particularly simple when the lattice is rectangular. In this case it takes the form:

$\begin{matrix} {{x = {\sum\limits_{n,m}{{x\left\lbrack {n,m} \right\rbrack}\;\delta\;\left( {{n\tau}_{r}/N} \right)\;\delta\;\left( {{mv}_{r}/M} \right)}}},} & (4.7) \end{matrix}$

B4.2 Transceiver structure Having specified the underlying structures and parameters we are now ready to describe the ZDMA modulation and de-modulation transforms. Given a periodic function x∈

(V/Λ) encoding the information bits, we define the modulated waveform φ_(tx)=M (x) through the rule:

$\begin{matrix} \begin{matrix} {{M(x)} = {Z_{{time},\epsilon}\left( {{\prod^{\epsilon}\left( w_{tx} \right)} \vartriangleright \left( {x \cdot P} \right)} \right)}} \\ {= {Z_{{time},\epsilon}\;\left( {w_{tx}\mspace{11mu}\left( {x \cdot P} \right)} \right)}} \end{matrix} & (4.8) \end{matrix}$

where Z_(time,ϵ); is the Zak transform converting between Zak and time domain waveforms, see (2.7). In words, the modulation first transforms the information function into a Zak waveform through multiplication by multiplication with P. Next, it shapes the bandwidth and duration of the waveform through twisted convolution with the two dimensional filter w. Last, it transforms the tamed Zak waveform into a time domain waveform through application of the Zak transform. To get a better understanding of the structure of the transmit waveform and the actual effect of two dimensional filtering we apply several algebraic manipulations to get more concrete expressions. First, we note that Z_(time,ϵ); is an intertwining transform thus obeying the relation: Z _(time,ϵ)(Π^(ϵ)(w _(tx))

(x·P))=Π^(time)(w _(tx))

Z _(time,ϵ)(x·P),  (4.9)

Second, assuming w_(tx)=w_(τ)

w_(ν), we can write Π^(time)(w_(τ)

w_(ν)) as the composition time Π^(time)(w_(τ))·Π^(time)(w_(ν)), thus expressing the two dimensional filtering operation as cascade of two consecutive one dimensional operations:

$\begin{matrix} {\begin{matrix} {{M(x)} = {{\prod^{time}\left( {w_{r}\mspace{11mu} w_{v}} \right)} \vartriangleright {Z_{{time},\epsilon}\left( {x \cdot P} \right)}}} \\ {= {{\prod^{time}\left( W_{t} \right)} \vartriangleright {\prod^{time}\left( w_{v} \right)} \vartriangleright {Z_{{time},\epsilon}\left( {x \cdot P} \right)}}} \\ {{= {w_{\tau}*\left\{ {W_{t} \cdot {Z_{{time},\epsilon}\left( {x \cdot P} \right)}} \right\}}},} \end{matrix}\quad} & (4.10) \end{matrix}$

where * stands for linear convolution on

and W_(t)=FT⁻¹(w_(ν)). We refer to the waveform Z_(time,ϵ)(x·P) as the bare ZDMA waveform. We see from Formula (4.10) that the transmit waveform is obtained from the bare waveform by applying the time window W_(t) followed by a convolution with the pulse w_(τ). In addition, one can verify that when x is sampled on the lattice Λ_(N,M)⊂V, see (4.6), the bare waveform is an infinite delta pulse train along the lattice: Λ_(N,M) ^(time) ={n/Ng ₁[1]+m/Mg ₂[1]:n,m∈

},  (4.11)

where the lattice Λ_(N,M) ^(time) is the projection of the lattice Λ_(N,M) on the delay axis. The projected lattice takes a particularly simple form when Λ=Λ_(r) is rectangular. In this case we have: Λ_(N,M) ^(time)=

τ_(r) /N,  (4.12)

We now proceed to describe the de-modulation mapping. Given a received waveform φ_(rx) we define its de-modulated image y=D(φ_(rx)) through the rule: D(φ_(rx))=Π^(ϵ)(w _(rx))

Z _(ϵ,time)(φ_(rx)),  (4.13)

We stress the fact that y is a Zak waveform (to be distinguished from a periodic function). We often incorporate as part of the de-modulation mapping another step of sampling y on the lattice Λ_(N,M), thus obtaining a sampled Zak waveform y_(s)∈H(Λ_(N,M),π_(ϵ)) which is a function on Λ_(N,M) satisfying the quasi periodicity condition: y _(s)(δ+λ)=ψ(β(δ,λ))π_(ϵ)(λ)⁻¹

y _(s)(δ),  (4.14)

To conclude, assuming x=x_(s)∈

(Λ_(N,M)/Λ) is a sampled periodic function, the ZDMA transceiver chain converts it into a sampled Zak waveform y_(s)∈H(Λ_(N,M),π_(ϵ)). Overall, the composition transform D·M is a linear transformation: D·M:

(Λ_(N,M)/Λ)→H(Λ_(N,M),π_(ϵ)),  (4.15)

taking sampled periodic functions to sampled Zak waveforms. In particular, we have that both the domain and range of D·M are finite dimensional vector spaces of dimension N·M. In the next subsection we will analyze in more detail the exact relation between the input variable x and the output variable y.

B4.3 Input output relation We first assume the channel between the transmitter and receiver is trivial. Furthermore, we assume the received filter is matched to the transmit filter, i.e., w_(rx)=w_(tx) ^(★). For simplicity we denote w=w_(tx) and assume w is decomposable, i.e., w=w_(τ)

w_(ν). At this stage we do not assume anything about the specific structure of the one dimensional filters w_(τ) and w_(ν). Given an input function x∈

(V/Λ), a direct computation reveals that y=D·M(x) is given by:

$\begin{matrix} {\begin{matrix} {y = {{\prod^{\epsilon}\left( w^{\bigstar} \right)} \vartriangleright {\prod^{\epsilon}{(w)}} \vartriangleright \left( {x \cdot P} \right)}} \\ {= {{\prod^{\epsilon}\left( {w^{\bigstar}\mspace{11mu} w} \right)} \vartriangleright \left( {x \cdot P} \right)}} \\ {{= {\left( {w^{\bigstar}\mspace{11mu} w} \right)\mspace{11mu}\left( {x \cdot P} \right)}},} \end{matrix}\quad} & (4.16) \end{matrix}$

So we see that y is given by the twisted convolution of x·P by the auto-correlation filter w^(★)

w. Our goal is to calculate an explicit formula for w^(★)

w. First we note that since w^(★)=w_(ν) ^(★)

w_(τ) ^(★), we can write:

$\begin{matrix} \begin{matrix} {{w^{\bigstar}\mspace{11mu} w} = {w_{v}^{\bigstar}\mspace{11mu} w_{\tau}^{\bigstar}\mspace{11mu} w_{\tau}\mspace{11mu} w_{v}}} \\ {{= {w_{v}^{\bigstar}\mspace{11mu} w_{\tau}^{(2)}\mspace{11mu} w_{v}}},} \end{matrix} & (4.17) \end{matrix}$

where w_(τ) ⁽²⁾=w_(τ) ^(★)=w_(τ) ^(★)*w_(τ) is the one dimensional auto-correlation function of the delay filter w_(t). In addition, since w_(τ) ⁽²⁾ is supported on the τ axis and w_(ν) ^(★) is supported on ν axis, we have the following simple relation:

$\begin{matrix} \begin{matrix} {{w_{v}^{\bigstar}\mspace{11mu}{w_{\tau}^{(2)}\left( {\tau,v} \right)}} = {{{w_{\tau}^{(2)}(\tau)} \otimes {\psi\left( {\tau\; v} \right)}}{w_{v}^{\bigstar}(v)}}} \\ {{= {{{w_{\tau}^{\bigstar}(\tau)} \otimes {M_{\tau}\left\lbrack w_{v}^{\bigstar} \right\rbrack}}(v)}},} \end{matrix} & (4.18) \end{matrix}$

Thus, for any given point (τ,ν), we can write w^(★)

(τ,ν) in the form: w ^(★)

w(τ,ν)=w _(τ) ⁽²⁾(τ)⊕⊗{tilde over (w)} _(ν) ⁽²⁾(ν),  (4.19)

where {tilde over (w)}_(ν) ⁽²⁾=M_(τ)[w_(ν) ^(★)]

w_(ν). We note that the definition of {tilde over (w)}_(ν) ⁽²⁾ depends on the point τ which is not apparent from the notation we use. Formula (4.19) is always true and is pivotal to get approximations of w^(★)

w under various assumptions on the filters w_(τ) and w_(ν). The case of interest for us is when w_(τ) and w_(ν) are square root Nyquist with respect to a bandwidth B and a duration T∈

^(≥0) respectively and, in addition, B·T>>1. In this case we can approximate {tilde over (w)}_(ν) ^(★)(ν)=ψ(τν)w_(ν) ^(★)(ν)˜w_(ν) ^(★)(ν), thus {tilde over (w)}_(ν) ⁽²⁾˜w_(ν) ⁽²⁾, which in turns imply that: w ^(★)

w˜w _(τ) ⁽²⁾

w _(ν) ⁽²⁾,  (4.20)

In particular, in the rectangular situation where Λ=Λ_(r), B=Nν_(r) and T=Mτ_(r) such that NM>>1, placing the QAM symbols on the lattice Λ_(N,M)=

(τ_(r)/N,0)⊕

(0,ν_(r)/M) yields:

$\begin{matrix} {{y = {{\prod^{\epsilon}\left( {w^{\bigstar}\mspace{11mu} w} \right)} \vartriangleright \left( {x \cdot P} \right) \sim {x \cdot P}}},} & (4.21) \end{matrix}$

thus allowing perfect reconstruction without equalization when the channel is AWGN. Next, we describe the input-output relation in the presence of a non-trivial channel H=Π^(time)(h) where h=h(τ,ν) is the delay Doppler impulse response. For the sake of the analysis, it is sufficient to assume that h is a single reflector, i.e., h(τ,ν)=δ(τ−τ₀,ν−ν₀)=δ(τ−τ₀)

δ(ν−ν₀). In the following computation we use the short notations δτ₀=δ(τ−τ₀) and δν₀=δ(ν−ν₀). Given an input x∈

(V/Λ), a direct computation reveals that the transmit-receive image y=D·H·M(x) is given by: y=Π ^(ϵ)(w★

h

w)

(x·P),  (4.22)

Our goal is to calculate an explicit formula for w^(★)

h

w. To do that, we first write:

$\begin{matrix} \begin{matrix} {{w^{\bigstar}\mspace{11mu} h\mspace{11mu} w} = {w_{v}^{\bigstar}\mspace{11mu} w_{\tau}^{\bigstar}\mspace{11mu}\delta_{\tau_{0}}\mspace{11mu}\delta_{v_{0}}\mspace{11mu} w_{\tau}\mspace{11mu} w_{v}}} \\ {= {w_{v}^{\bigstar}\mspace{11mu}{L_{\tau_{0}}\left\lbrack w_{\tau}^{\bigstar} \right\rbrack}\mspace{11mu}{M_{v_{0}}\left\lbrack w_{\tau} \right\rbrack}\mspace{11mu}{L_{v_{0}}\left\lbrack w_{v} \right\rbrack}}} \\ {{= {w_{v}^{\bigstar}\mspace{11mu}{L_{\tau_{0}}\left\lbrack {\overset{\sim}{w}}_{\tau}^{(2)} \right\rbrack}\mspace{11mu}{L_{v_{0}}\left\lbrack w_{v} \right\rbrack}}},} \end{matrix} & (4.23) \end{matrix}$

where {tilde over (w)}_(τ) ⁽²⁾=w_(τ) ^(★)

M_(ν0)[w_(τ)]. In addition, we have:

$\begin{matrix} \begin{matrix} {{w_{v}^{\bigstar}\mspace{11mu}{L_{\tau_{0}}\left\lbrack {\overset{\sim}{w}}_{\tau}^{(2)} \right\rbrack}\left( {\tau,v} \right)} = {{L_{\tau_{0}}\left\lbrack {\overset{\sim}{w}}_{\tau}^{(2)} \right\rbrack}{(\tau) \otimes {\psi\left( {\tau\; v} \right)}}{w_{v}^{\bigstar}(v)}}} \\ {{= {{L_{T_{0}}\left\lbrack {\overset{\sim}{w}}_{\tau}^{(2)} \right\rbrack}{(\tau) \otimes {M_{\tau}\left\lbrack w_{v}^{\bigstar} \right\rbrack}}(v)}},} \end{matrix} & (4.24) \end{matrix}$

Hence, overall we can write: w ^(★)

h

w(τ,ν)=L _(τ) ₀ [{tilde over (w)} _(τ) ⁽²⁾](τ)⊗L _(ν) ₀ [{tilde over (w)} _(ν) ⁽²⁾](ν),  (4.25)

where {tilde over (w)}_(ν) ⁽²⁾=M_(T)[w_(ν) ^(★)]

w_(ν). If we assume that w_(τ) and w_(ν) are Nyquist with respect to a bandwidth B and a duration T respectively, and, in addition have B·T>>1 and ν₀<<B then we can approximate {tilde over (w)}_(τ) ⁽²⁾˜w_(τ) ⁽²⁾ and {tilde over (w)}_(ν) ⁽²⁾˜w_(τ) ⁽²⁾, which, in turns, imply: w ^(★)

h

w=h*w _(τ) ⁽²⁾ *w _(ν) ⁽²⁾,  (4.26)

B4.4 Channel acquisition Looking back at the input output relation y=h_(w)

(x·P) where h_(w)=w^(★)

h

w, we proceed to derive a simple acquisition scheme of the filtered channel impulse response h_(w). To this end, we fix a point ν₀∈V and consider the standard pulse structure P(ag₁+bg₂)=1 for 0≤a,b≤1. Given these choices, we define the pilot structure as the Zak waveform φ.

B4.5 Weak ZDMA In this subsection, we describe a weak variant of the ZDMA transceiver that can be architected as a pre-processing layer over multi-carrier transceiver. We refer to this transceiver as w-ZDMA. The definition of the w-ZDMA transceiver depends on similar parameters as the ZDMA transceiver we described before however, with few additional assumptions. First assumption is that the transmit and receive filters are periodic with period Λ, i.e., w_(tx), w_(rx)∈

(V/Λ). in other words, w_(tx/rx)=SF(W_(tx/rx)) where w_(tx), w_(rx)∈

(V/Λ) are discrete window functions and SF is the symplectic Fourier transform. The support of the window W_(tx) determines the bandwidth and duration of the transmission packet. Typically, we take the receive filter to be matched W_(rx)=W _(tx) or, equivalently, that w_(rx)=w_(tx) ^(★). Another assumption is that the generator signal P∈H(V,π_(ϵ)) satisfy the orthogonality condition: P·P=1,  (4.27)

We note that condition (4.27) is equivalent to Gabor orthogonality condition of the waveform p=Z_(time,ϵ)(P) with respect to the Heisenberg lattice Im {circumflex over (ϵ)}={(λ,ϵ(λ)):λϵΛ}. To see this, let λ∈Λ and consider the inner product

p,π^(time)(λ,ϵ(λ))

p

.

Now write:

$\begin{matrix} \begin{matrix} {\left\langle {p,{{\pi^{time}\left( {\lambda,{\epsilon(\lambda)}} \right)} \vartriangleright p}} \right\rangle = \left\langle {{Z_{{time},\epsilon}(P)},{{\pi^{time}\left( {\lambda,{\epsilon(\lambda)}} \right)} \vartriangleright {Z_{{time},\epsilon}(P)}}} \right\rangle} \\ {= \left\langle {{Z_{{time},\epsilon}(P)},{Z_{{time},\epsilon}\left( {{\pi^{\epsilon}\left( {\lambda,{\epsilon(\lambda)}} \right)} \vartriangleright P} \right)}} \right\rangle_{t}} \\ {= {\int_{V/A}{{\overset{\_}{P}(v)}\left\{ {{\pi^{\epsilon}\left( {\lambda,{\epsilon(\lambda)}} \right)} \vartriangleright P} \right\}\;(v)\;{dv}}}} \\ {= {\int_{V/A}{{\psi\left( {\omega\left( {\lambda,v} \right)} \right)}{\overset{\_}{P}(v)}{P(v)}{dv}}}} \\ {= {\int_{V/A}{{{\psi\left( {\omega\left( {\lambda,v} \right)} \right)} \cdot 1}\;{dv}}}} \\ {= \left\{ {\begin{matrix} 1 & {\lambda = 0} \\ 0 & {\lambda \neq 0} \end{matrix},} \right.} \end{matrix} & (4.28) \end{matrix}$

where in the second equality we use the fact Z_(time,ϵ) is an intertwining transform and in the fourth equality we use (2.5). Given an input function x∈

(V/Λ) encoding the information bits, we shape the band and duration of the transmitted waveform through periodic convolution with w_(tx), i.e., x

w_(tx)*x. Overall, the signal x is modulated according to the following rule: M(x)=Z _(time,ϵ)((w _(tx) *x)·P),  (4.29)

It is illuminating to compare Formula (4.29) with Formula (4.8). One observes that the main difference is in the way the shaping filter is applied where in ZDMA it is applied through the operation of twisted convolution x

w_(tx)

(x·P) and in w-ZDMA through the operation of periodic convolution x

(w_(tx)*x)·P. We next explain how the modulation rule (4.29) can be expressed as a layer over multi-carrier modulation scheme. To this end, let us write w*x in the form w*x=SF(W·X) where x=SF (X), that is:

$\begin{matrix} {{{w*{x(v)}} = {\sum\limits_{\lambda \in A}{{\psi\left( {- {\omega\left( {v,\lambda} \right)}} \right)}{{W(\lambda)} \cdot {X(\lambda)}}}}},} & (4.30) \end{matrix}$

For every ν∈V. Hence:

$\begin{matrix} {\begin{matrix} {{Z_{{time},\epsilon}\left( {\left( {w*x} \right) \cdot P} \right)} = {\sum\limits_{\lambda \in A}{{{W(\lambda)} \cdot {X(\lambda)}}\;{Z_{{time},\epsilon}\left( {{\psi\left( {- {\omega\left( {v,\lambda} \right)}} \right)}P} \right)}}}} \\ {= {\sum\limits_{\lambda \in A}{{{W(\lambda)} \cdot {X(\lambda)}}\;{Z_{{time},\epsilon}\left( {{\psi\left( {\omega\left( {\lambda,v} \right)} \right)}P} \right)}}}} \\ {= {\sum\limits_{\lambda \in A}{{{W(\lambda)} \cdot {X(\lambda)}}\;{Z_{{time},\epsilon}\left( {{\pi^{\epsilon}\left( {\lambda,{\epsilon(\lambda)}} \right)} \vartriangleright P} \right)}}}} \\ {= {{\sum\limits_{\lambda \in A}{{{W(\lambda)} \cdot X}(\lambda){\pi^{time}\left( {\lambda,{\epsilon(\lambda)}} \right)}}} \vartriangleright {Z_{{time},\epsilon}(P)}}} \\ {= {{\sum\limits_{\lambda \in A}{{{W(\lambda)} \cdot X}(\lambda){\pi^{time}\left( {\lambda,{\epsilon(\lambda)}} \right)}}} \vartriangleright p}} \\ {{= {{\prod^{time}\left( {{\hat{\epsilon}}_{pf}\left( {W \cdot {{SF}(x)}} \right)} \right)} \vartriangleright p}},} \end{matrix}\quad} & (4.31) \end{matrix}$

In case the Heisenberg character ϵ=1, e.g., when Λ=Λ_(r) is rectangular, the modulation formula becomes M(x)=Π^(time)(W·SF(x))

p. We now proceed to describe the de-modulation rule implemented at the receiver side. Given a time domain waveform φ_(rx), the receiver demodulates it according to the following formula: D(φ_(rx))=Z _(ϵ,time)(φ_(rx))· P,  (4.32)

Observe that when the channel is identity, due to the orthogonality condition (4.27), we obtain perfect reconstruction after composing modulation and demodulation:

$\begin{matrix} {\begin{matrix} {{D \circ {M(x)}} = {{x \cdot P}\overset{\_}{P}}} \\ {= {x \cdot 1}} \\ {{= x},} \end{matrix}\quad} & (4.33) \end{matrix}$

We further note that the orthogonality condition is not essential for achieving perfect reconstruction. In fact, one needs to impose is that P is non-degenerate, that is, that P P is nowhere vanishing. For non-degenerate P one can reconstruct the input x as: x=D·M(x)/PP,

The use of general non-degenerate generator functions give rise to non-orthogonal variants of the w-ZDMA transceiver. For a non-trivial channel transformation of the form H=π^(time)(ν₀) where ν₀=(τ₀,ν₀) we get:

$\begin{matrix} {\begin{matrix} {y = {{{D \circ H \circ \; M}\;(x)} = {D\;\left( {{\pi^{time}\left( v_{0} \right)} \vartriangleright {M\;(x)}} \right)}}} \\ {= {Z_{\epsilon,{time}}\;{\left( {{\pi^{time}\left( v_{0} \right)} \vartriangleright {M\;(x)}} \right) \cdot \overset{\_}{P}}}} \\ {= {\left\{ {{\pi^{\epsilon}\left( v_{0} \right)} \vartriangleright {Z_{\epsilon,{time}}\left( {M\;(x)} \right)}} \right\} \cdot \overset{\_}{P}}} \\ {{= {\left\{ {{\pi^{\epsilon}\left( v_{0} \right)} \vartriangleright {x \cdot P}} \right\}\overset{\_}{P}}},} \end{matrix}\quad} & (4.34) \end{matrix}$

where the second equality is the definition of D, the third equality is the intertwining property of the Zak transform and the last equality is the definition of M. Finally, let ν=(τ,ν) where 0≤τ,ν<1 and evaluate y at τ: y(ν)=ψ(−β(ν₀,ν₀))ψ(β(ν₀,ν))x(ν−ν₀)P(ν−ν₀) P (ν)  (4.35)

Assuming ν₀ is small compared to the lattice dimensions and that P is a continuous function we get the approximation (The continuity assumption of G does not always hold, for example in the most common case when G=Z⁻¹(1_([0,T]))—the case of the standard window function.)

$\begin{matrix} {\begin{matrix} {{y(v)} \simeq {{x\left( {v - v_{0}} \right)}\; P\;(v)\;\overset{\_}{P}\;(v)}} \\ {{= {x\left( {v - v_{0}} \right)}}\;,} \end{matrix}\quad} & (4.36) \end{matrix}$

where for the approximation we used the fact that ψ(β(ν₀,ν)), ψ(−β(ν₀,ν₀))≅1 and that P(ν−ν₀)≅P(ν) by continuity. Note that when P corresponds to the standard window (see Example 2.2) the approximation (4.36) is no longer valid since P is not continuous at the boundaries of the fundamental cell. We leave it to the reader to compute the appropriate approximation in this case.

FIG. 11 is a flowchart representation of an example of a wireless communication method 1100. The method 1100 may be implemented at a transmitter-side in a wireless communication system. The method 1100 includes encoding (1102) information bits as a periodic sequence of quadrature amplitude modulation (QAM) symbols. For example, the periodic sequence may be periodic in the symbol period domain. For example, the perioidicity may also be exhibited along the Doppler dimension in a delay-Doppler domain representation of the QAM-modulated data bits. The method 1100 includes convolving (1104) the periodic sequence with a periodic pulse function, thereby generating a filtered periodic sequence. For example, as described in Section A.0, the convolution operation may be mathematically represented as Fourier Transform or as a 1-D convolution with the sinc function. The method 1100 includes transforming (1106) the filtered periodic sequence to a delay-Doppler domain waveform, converting (1108) the delay-Doppler domain waveform to a time domain waveform; and transmitting (1110) the time domain waveform.

As described in the present document, the transforming operation 1106 may be performed by applying a Zak transform to the filtered periodic sequence. In some embodiments, the periodic pulse functions used in operation 1104 may include Dirichlet sinc function. Some examples are further described in Section A4.2 and with respect to FIG. 7 and FIG. 9. In some embodiments, a tapering window may be applied for the convolution operation. Some embodiments are described in Section A4.2.

In some embodiments, for example, as described in Section A.2, the encoding information bits as a periodic sequence incudes encoding the information bits as a two-dimensional periodic sequence having a periodicity of N in a first dimension and M in a second dimension, where N and M are integers.

Accordingly, in some embodiments, the time domain waveform as described with respect to operation 1110 may be transmitted over a wireless channel. The waveform may include a sequence of modulated QAM symbols, without having to add cyclic prefix. The time domain waveform may correspond to a mathematical equivalent of the results of operations 1102 to 1108. For example, in some embodiments, the generation operation 1104 may be followed by conversion to time domain waveform, without going through an intermediate delay-Doppler stage.

FIG. 12 is a flowchart representation of another example of a wireless communication method 1200. The method 1200 can be implemented by a wireless transmitter. The method 1200 includes transforming (1202) an information signal to a discrete lattice domain signal, shaping (1204) bandwidth and duration of the discrete lattice domain signal by a two-dimensional filtering procedure to generate a filtered information signal, generating (1206) a time domain signal from the filtered information signal, and transmitting (1208) the time domain signal over a wireless communication channel.

For example, in some implementations of the method 1200, the discrete lattice domain includes a Zak domain. As described further in Sections A and B, the two-dimensional filtering procedure includes a twisted convolution with a pulse. In some embodiments, the pulse is a separable function of each dimension of the two-dimensional filtering. Some example implementations are described in Sections A0, A3.

The method 1200 may be implemented by a transmitter device for transforming modulated signals that use QAM (or quadrature phase shift keyed) constellations and are generated using 2D filtered versions of a lattice domain signal. Notably, no cyclic prefixes are inserted within the periodic signal, thereby saving any overheads of transmission that are wasted in the conventional OFDM techniques. As previously discussed, the filtered periodic sequences, generated during the method 1100 also similarly eschew the use of cyclic prefixes, thereby saving overhead of traditional OFDM communication.

FIG. 13 shows an example of a wireless transceiver apparatus 1300. The apparatus 700 may be used to implement various techniques described herein. The apparatus 1300 includes a processor 1302, a memory 1304 that stores processor-executable instructions and data during computations performed by the processor. The apparatus 1300 includes reception and/or transmission circuitry 1306, e.g., including radio frequency operations for receiving or transmitting signal and/or receiving data or information bits for transmission over a wireless network.

It will be appreciated that techniques for data modulation are disclosed in which information signal can be transmitted using multiple QAM subcarriers without using a cyclic prefix. In some embodiments, a modulation technique, called OFDM-MultiCarrier (MC) may be used in which QAM symbols are convolved with a periodic pulse function. In some embodiments, a Zak domain representation of a signal is used for shaping bandwidth and duration of a modulated information signal.

The disclosed and other embodiments, modules and the functional operations described in this document can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this document and their structural equivalents, or in combinations of one or more of them. The disclosed and other embodiments can be implemented as one or more computer program products, i.e., one or more modules of computer program instructions encoded on a computer readable medium for execution by, or to control the operation of, data processing apparatus. The computer readable medium can be a machine-readable storage device, a machine-readable storage substrate, a memory device, a composition of matter effecting a machine-readable propagated signal, or a combination of one or more them. The term “data processing apparatus” encompasses all apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers. The apparatus can include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. A propagated signal is an artificially generated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode information for transmission to suitable receiver apparatus.

A computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a standalone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network.

The processes and logic flows described in this document can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit).

Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read only memory or a random access memory or both. The essential elements of a computer are a processor for performing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks. However, a computer need not have such devices. Computer readable media suitable for storing computer program instructions and data include all forms of non-volatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto optical disks; and CD ROM and DVD-ROM disks. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry.

While this patent document contains many specifics, these should not be construed as limitations on the scope of an invention that is claimed or of what may be claimed, but rather as descriptions of features specific to particular embodiments. Certain features that are described in this document in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or a variation of a sub-combination. Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results.

Only a few examples and implementations are disclosed. Variations, modifications, and enhancements to the described examples and implementations and other implementations can be made based on what is disclosed. 

What is claimed is:
 1. A wireless communication method, comprising: transforming an information signal to a discrete lattice domain signal; shaping bandwidth and duration of the discrete lattice domain signal by a two-dimensional filtering procedure to generate a filtered information signal, wherein the two-dimensional filtering procedure includes a twisted convolution with a pulse; generating a time domain signal from the filtered information signal; and transmitting the time domain signal over a wireless communication channel.
 2. The method of claim 1, wherein the discrete lattice domain includes a Zak domain.
 3. The method of claim 1, wherein the pulse is a separable function of each dimension of the two-dimensional filtering.
 4. The method of claim 1, wherein the time domain signal comprises modulated information signal without an intervening cyclic prefix.
 5. The method of claim 4, wherein the modulated information signal uses quadrature amplitude modulation (QAM) symbols.
 6. The method of claim 4, wherein the modulated information signal is based on an orthogonal time frequency space (OTFS) modulation or an OTFS multicarrier (OTFS-MC) modulation.
 7. The method of claim 1, wherein the pulse is a sinc function.
 8. A wireless communication method, implementable by a wireless communication apparatus, comprising: encoding information bits as a periodic sequence of quadrature amplitude modulation (QAM) symbols; convolving the periodic sequence with a periodic pulse function, thereby generating a filtered periodic sequence; transforming the filtered periodic sequence into a delay-Doppler domain waveform; converting the delay-Doppler domain waveform to a time domain waveform; and transmitting the time domain waveform.
 9. The method of claim 8 wherein the transforming comprises applying a Zak transform.
 10. The method of claim 8, wherein the periodic pulse function comprises a Dirichlet sinc function.
 11. The method of claim 8, wherein the convolving the periodic pulse function comprises applying a tapering window function.
 12. The method of claim 8, wherein the encoding information bits as a periodic sequence includes encoding the information bits as a two-dimensional periodic sequence having a periodicity of N in a first dimension and M in a second dimension, where N and M are integers.
 13. A wireless signal transmission apparatus comprising a processor, wherein the processor is configured to implement a method comprising: transforming an information signal to a discrete lattice domain signal; shaping bandwidth and duration of the discrete lattice domain signal by a two-dimensional filtering procedure to generate a filtered information signal, wherein the two-dimensional filtering procedure includes a twisted convolution with a pulse; generating a time domain signal from the filtered information signal; and transmitting, using transmission circuitry, the time domain signal over a wireless communication channel.
 14. The apparatus of claim 13, wherein the discrete lattice domain includes a Zak domain.
 15. The apparatus of claim 13, wherein the pulse is a separable function of each dimension of the two-dimensional filtering.
 16. The apparatus of claim 13, wherein the time domain signal comprises modulated information signal without an intervening cyclic prefix.
 17. The apparatus of claim 16, wherein the modulated information signal uses quadrature amplitude modulation (QAM) symbols.
 18. The apparatus of claim 16, wherein the modulated information signal is based on an orthogonal time frequency space (OTFS) modulation or an OTFS multicarrier (OTFS-MC) modulation.
 19. The apparatus of claim 13, wherein the twisted convolution between a first function h₁ and a second function h₂ is defined as: ${{h_{1^{*}\sigma}{h_{2}(\upsilon)}} = {\int\limits_{{\upsilon_{1} + \upsilon_{2}} = \upsilon}{{\exp\left( {j\; 2\;{{\pi\beta}\left( {\upsilon_{1},\upsilon_{2}} \right)}} \right)}{h_{1}\left( \upsilon_{1} \right)}{h_{2}\left( \upsilon_{2} \right)}}}},$ wherein β is a polarization form.
 20. The apparatus of claim 13, wherein the pulse is a sinc function. 